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Examination  Manuals 


No.    II. 


4^GEBB^ 


-^-O-H^ 


IN  MEMOMAM 
Irving  Stringham 


n 


WENTWORTH  &  HILL'S 


EXAMINATION  MANUALS. 


No.  II. 


ALGEBRA. 


BOSTON: 

GINN,   HEATH,   &  COMPANY. 

1884. 


Entered,  according  to  Act  of  Congress,  in  the  year  1883, 

By  GEORGE  A.  WENTWORTH  and  GEORGE  A.  HILL, 

In  the  Office  of  the  Librarian  of  Congress  at  Washington. 


J.  S.  Cushing  &  Co.,  Printers,  Boston. 


L-% 


PREFACE. 


THIS  Manual  consists  of  two  parts :  The  first  part  contains  one 
hundred  and  fifty  examination  papers,  the  questions  for 
which  have  been  selected  mainly  from  the  best  English,  French, 
and  German  collections  of  problems.  These  papers  may  be  divided 
into  three  groups.  The  first  fifty  papers  embrace  the  subjects 
of  Elementary  Algebra  as  far  as  Quadratic  Equations;  the  next 
fifty  papers  also  include  Quadratic  Equations  and  Radical  Expres- 
sions; the  last  fifty  papers  extend  over  still  more  ground,  includ- 
ing several  topics  usually  regarded  as  belonging  to  Higher 
Algebra. 

In  each  of  these  groups  the  earlier  papers  will  be  found 
somewhat  easier  than  the  later  ones.  The  papers  are  intended 
to  be  hour  papers,  but  if  any  of  them  are  thought  to  be  too 
long  for  one  hour,  the  time  may  be  increased  or  the  length  of 
the   paper  reduced  by  omitting  one   or  more    of  the   questions. 

The  second  part  of  the  Manual  is  a  collection  of  recent 
papers  actually  set  in  various  American  and  English  institutions 
of  learning. 

There   are   two  ways   in   which   the   Manual   may  be   used. 

First :  To  test  the  learner's  knowledge  in  the  usual  way  by 
means  of  an  examination.  For  this  purpose  the  class  will  come 
to  the  recitation-room  provided  with  the  Manual  and  blank 
books,  and  the  teacher  will  simply  designate  by  number  the 
paper   to   be  worked. 

Secondly :  To  exercise  the  learner  from  day  to  day  in  the 
various  rules   and  processes,  to  detect  his  weak  points,  and  ascer- 


800575 


IV  PREFACE. 


tain  where  he  most  needs  assistance.  This  may  be  done  by- 
assigning  exercises  to  be  worked  in  the  class-room,  or  by  assign- 
ing to  each  member  of  the  class  a  paper  with  directions  to 
hand  in  the  solutions,  neatly  worked  out,  at  a  subsequent  reci- 
tation. 

The  Manual  will  be  found  especially  useful  in  reviewing  the 
subject   of  Algebra,  and  in   preparing   for   examinations. 

Answers  to  the  problems  in  the  first  one  hundred  and  fifty 
papers,  bound  separately  in  paper  covers,  can  be  had  by  teachers 
only,  on   application   to   the  publishers. 

G.   A.   WENTWORTH. 
G.  A.  HILL 


SPECIMEN  PAPER  WORKED  OUT. 


1.    Simplify 


a _  [26  +  {3  c  -  3a  -  (a  +  b)}  +  {2a  -  (b  +  c)}] 

=  a-[26+{3c-3a-a-&}+{2a-6-c}] 

=  a  — [26  + 3c -3a  — a -6  + 2a  — 6 -c] 

=  a-26  — 3c  +  3a  +  a  +  6  — 2a  +  6  +  c 

=  3a-2c. 

Ans.   3  a  —  2  c. 


2.   Resolve  into  factors 

xi  _  ^2  +  22  _  a2  _  2X2  +  2ay 

=  (a;2  -  2xz  +  z2)  -  (a2  -  2  ay  +  y2) 
-  (s  -  z)2  -  (a  -  3/)2 
=  {(ir-2)-(a-2/)}{(a;-2)  +  (a-y)} 
=  (a;  —  2  -  a  +  y)  (x  —  z  +  a  -  3/). 


3.   Find  the  H.O.F.  of 


5a;2  (12a8 +  4^  +  17a  -3) 
12a;3  +  4a;2  +  17s  -3 
12a3  +  4a;2-      a; 


5x2(12ar»  +  4a;2+17a;-3) 

and  10a;  (24ar»  -  52a;2  +  14 ar  - 1). 

10a:(24arJ-52a;2  +  14a;-l) 
24a*  -  52a;2  +  14a; -1 
24x3+    Sx2  +  34z-6 


3)18  a; -3 
6*-l 


-5)- 

-60  a;2- 

20  a; + 

5 

12a;2 + 
12  a;2- 

4a;- 
2x 

1 

6a:- 
6x- 

1 

1 

Reserve  5  a? 

2 


x 

2z  +  l 


Ans.   5x(Qx  —  l). 


VI  EXAMINATION    MANUAL. 


4. 

Simplify 

r     J      3               7          4 -20a; 

l-2x      l  +  2a;      4a;2 -1 

3               7          4 -20a; 

l-2a?      l  +  2a;      4a;2 -1 

3               7          4-20a; 

l-2ar      1+2sb      l-4a;2' 

L.C.D.  =  l-4a;2. 

3  +    6  x  =  first  numerator, 

—  7  +  14  a;  =  second  numerator, 

4  —  20  a;  =  third  numerator. 

0  =  sum  of  numerators. 

Arts.   0. 

5, 

Solve       10a; +  152/ -242=     41 
15s-12y  +  16«-      10 
18a; -14y-    7s  — —13. 

(1) 

(2) 

■       (3) 

Multiply  (1)  by  2,                       20  x  +  30 y  -  48  z  = 
Multiply  (2)  by  3,                      45  x  -  36 y  +  48  z  = 

82 
30 

Add,                                            65a;-    62/ 

112 

(4) 

Multiply  (2)  by  7,                 105  a;  -    84  y  +  112  2  = 
Multiply  (3)  by  16,               288  x  -  224y  -  1122  = 

70 

-208 

Add,                                                   393  x  - 

308  y  = 

-138 

(5) 

Multiply  (4)  by  154,                      10010  a;  - 
Multiply  (5)  by  3,                            1179  a;- 

924y  = 
924y  = 

17248 
-414 

Subtract,                                           8831  x 

= 

17662 
2< 

Substitute  value  of  x  in  (4),                    130 

.  *.  x  — 

-6y  = 

112. 

.-.y  =  3. 
Substitute  values  of  x  and  y  in  (1),  20  +  45  —  24  2  =  41. 

.-.  2=1. 
Ans.   a;  =  2,  y  =  3,  2  =  1. 

6,  A  passenger  train,  after  travelling  an  hour,  is  detained 
15  minutes;  after  which  it  proceeds  at  three-fourths 
of  its  former  rate,  and  arrives  24  minutes  late.  If 
the  detention  had  taken  place  5  miles  farther  on, 
the  train  would  have  been  only  21  minutes  late. 
Determine  the  usual  rate  of  the  train. 


SPECIMEN   PAPER   WORKED    OUT.  Vll 

Let  x  =  usual  rate  of  train  per  hour, 

and  y  =  number  of  miles  train  has  to  run. 

Then  y  —  x  =  number  of  miles  train  has  to  run  after  detention, 

y  ~x  —  number  of  hours  usually  required  to  run  y  —  x  miles. 

and  ^-r —  =  number  of  hours  actually  required  to  run  y  —  x  miles, 

Since  the  detention  was  15  minutes,  and  the  train  was  24  minutes 
late,  the  loss  in  running-time  is  9  minutes,  or  ^  of  an  hour. 

•  V~x  ..    V~x  -  3  m 

'fa?  a;         20  K  ' 

If  the  detention  had  occurred  5  miles  farther  on,  the  loss  in  run- 
ning-time would  have  been  6  minutes,  or  ^  of  an  hour. 

.   y-x-5     y-x-5_l  f~ 


Simplify  (1), 
Simplify  (2), 

Subtract  (1)  from  (2), 

.    Solve 

1         + 

2a;2  +  a;  +  l      2ar 

1 
(2a;-l)(a;  +  l)      (2a;- 

Simplify,      abx  —  ab  H 

U2x2-a2 

\x                x            10                 K  J 

202/- 29a;  =     0 
20y-26a;=100 

3a;  =  100 
.-.  a;  =  33J. 

Ans.   33|  miles, 

1                    a           2  bx  +  b 

!  —  3  a;  +  1      2  bx  —  b      ax2  —  a 
1                        a                  2bx  +  b 

l)(a;-l)      b(2x-l)      a(sc-l)(aj  +  l) 

L.C.D.  =  ab(x-  l)(x  +  l)(2a;  -  1). 
-  abx  +  ab  =  a2x2-a2-4:b2x2  +  b2, 
2abx  =  a2x2-a2-4cb2x2  +  b2, 
x2  +  2abx  =  b2-a2, 

(U2 -  a2)x2  +  2abx  =  b2-  a2. 
Complete  the  square,  multiplying  by  4  times  the  coefficient  of 
and  adding  the  square  of  the  coefficient  of  x, 

4(462- a2)2x2  +  ( )  +  (2ab)2  =  16¥-I6b2a2  +  4a4,    • 
(4Z>2  -  a2)  2x  +  2ab  =  ±  (4  b2  -  2a2), 
(4&2 


2b  +  a 


EXAMINATION    MANUAL. 


8.    Solve 

a;  +  y  =  4 

(1) 

x*  +  2/*  =  82 

(2) 

Put  u  +  v  for  x,  and  u  —  v 

for  y. 

(1)  becomes 

2u  =  4. 
.-.  u  =  2. 

(2)  becomes 

w4+16w2v2+'y*  =  41 

(3) 

Substitute  2  for  u  in  (3), 

16  +  24v2  +  ^=41, 
t^  +  24v2  =  25, 
^  +  ()  +  144  =  169, 
a2  +  12  =  ±  13, 

v2  =  1  or 

-25, 

a  =  ±  1  or  ±  V^ 

-25. 

(  a;  =  3,  1, 

or  2± 

V^25, 

^    iy-1,3, 

or  2  + 

V-25. 

9.    Show  that  2a/«362,  -^o^  2-vl^  are  similar  surds. 

2^52  =  2a^62", 
y/a9>=2bW, 

Since  they  all  have  the  game  surd  factor  they  are  similar  surds. 


10.    Simplify 


=  (2ab)$x(3ab2)*  +  (5ab*)l 
=  V{2^bfx^(Mb2J2  -  y/5ab3 


V23  X  32  X  a5  X  V  -r- 


23x32a567 
5a&3 
«/23x32x55a46* 


r>« 


=  |V225000a*64 


iV225000a464. 


SPECIMEN    PAPER    WORKED    OUT.  ix 

11.  Expand  f—  -  </f 

(  2x2y-1-y$f=(2x2y-1f-6(2x2y-1Y(yt)  +  15  (2  as*  y"1)*^)* 
-  20(2x2y-1)3(y$f  +  15(2x23/"1)2(y^ 
-6(2x*y-i)(yi?  +  (yty 
-  64z12y-6  -  192a;10 y-*  +  240cc8y-§-  lBO^y'1 
+  60rr*3/^-12x22/5  +  t/*. 

Am.   64:X12y-6-192x10y-^}etc. 

12.  Find  the  value  of 

j/3.1416  x  4771.21  x  2.7183* 
30.103*  X  0.4343*  X  69.897*' 

log       3.1416=  0.4971  =0.4971 

log  4771.21     -  3.6786  =3.6786 

£log       2.7183=  0.4343^2  =0.2172 

4colog      30.103    =  (8.5214  -  10)  X  4  =  4.0856  -  10 

£colog       0.4343=  0.3622-^2  =0.1811 

4colog     69.897   =  (8.1555  -  10)  x  4  =  2.6220- 10 

11.2816-20 
30  -30 


5 )  41.2816 -50 


i|41 


8.2563  -  10 

Am.  0.01804. 


EXAMINATION  MANUAL. 


ALGEBRA. 


1. 

1.  Ifa=l,  5  =  3,  c  =  5,d=0t 

find  the  value  of  a2  +  2  b2  +  3  c2  +  4  rf2. 

2.  Add  a  + 6-2c,   8a  +  4e  +  26,   3c-2a-65, 

and  25  -  2c  +  3a. 

3.  Multiply  l  +  x  +  x4  +  x5  by  1  —  a? -f  a*  —  s*. 

4.  Resolve  into  the  simplest  factors  a2x2y  —  b2xy2. 

5.  Find  the  L.C.M.  of  xy,  x  —  y,  and  y3  —  x2y. 

o    «■      i7    7#  — 10      Sx—7     27x  —  30    . 

6.  Simplify — 

r    J         5  6  30 

7.  Solve   2(a?-3)  +  6(a?-2)  =  54. 

8.  Divide  a  line  12  inches  long  into  two  parts,  such  that 

the  greater  shall  be  five  times  the  less. 

2. 

1.  Ifa  =  l,  b  =  -l,  c  =  0,  d=2, 

find  the  value  of  (ac  —  bd)  (ad— be)  (ab  —  cd). 

2.  Divide  a2  +  ab  +  2a<?  -  2b2  +  7fo  -  3c2  by  a-  5  +  3c. 

3.  Resolve  into  the  simplest  factors,  a2x2  —  a2y2. 

4.  Find  the  H.O.F.  of  9*2  -  25  and  9*2  +  3a;  -  20. 

5.  Add^,    *=£,    and5"* 


ab  ac  be 

6.  Solve  5 -  3(4 -x)  +  4(3- 2x)  =  0. 

7.  A  and  B  together  have  $4900.      A  invests  £  of  his 

money  in  business,  and  B  \  of  his,  and  each  has  the 
same  sum  remaining.     How  much  money  had  each  ? 

8.  Solve  2o;-f  3y  =  221 

3s  +  4y  =  31J 


EXAMINATION   MANUAL. 


3. 

1.   If  a  =  l,  &  =  0,  <?  =  -!, 

evaluate  (1  —  ab)  (1  —  ac)  (1  —  be). 


2.  "Write  down  the  product  of  x  +  ->  a?  —  ^>  x2-}-—- 

3.  Simplify  3a- J[5+{2o-(5-a;)}]. 

4.  Find  the  H.C.F.  of  6 a6 -  6 a3 a;3  and  8a6-8a6#. 

5.  Subtract  — — —  from 


ax  ax  —  x2 

6.  Solve   \x+\(x  —  l)  =  a?-4. 

7.  Nine  years  ago  A  was  three  times  as  old  as  B,  but  now 

he  is  only  twice  as  old.     Find  the  present  ages  of 
A  and  B. 

8.  Solve  *  +  1  =  l 

3     2     6 

7  3      21 

4. 

1.  Multiply  a  —  b-\-chy  a  +  b  —  c,  and  find  the  value  of 

the  product  when  a  =  9,  b  =  4,  c  =  3. 

2.  From  5a  +  3c -45  -  7a7- e 

take  4a  +  7c7+5e  — 55— 6c. 

3.  Divide  xG  —  y6  by  a;  —  y. 

4.  Solve  ^-U*  =  l£-(^  +  5). 

8  12        ?      \       •' 

5.  Solve   7x+   4y  =  17yl 

6a;-10y  =    8    J 

6.  Besolve  into  factors  y2  +  25  y  —  150. 

7.  Simplify    a'~4  Xft2~25. 

^    J  a2+5a      a2  +  2a 

8.  Find  a  number  such  that  the  sum  of  its  fifth  and  its 

seventh   parts   shall    exceed   the    difference    of    its 
fourth  and  its  seventh  parts  by  99. 


ALGEBRA. 


5. 

1.  Simplify  3 0  +  a)  (y  -  b)  -  {  -  aty  -  (c  -d)]\. 

2.  Find  the  H.O.F.  of  12*4  -  108a;2  and  16#4-  48*3. 

3.  Find  the  L.C.M.  of  4  ab  (a2  -  b2)3  and  6  a3  b2  -  6  a2b3. 

4.  Multiply-—^- by 


x2-2x-S     J   x2  +  2x-Z 

5.  Solve    1+JL_JL  =  L 

a;     2x      ox      3 

6.  What  sum  is  that  which  is  as  much  greater  than  $20  as 

its  half  is  less  than  $20? 

7.  Two  trains  travelling,  one  at  26  and  the  other  at  30 

miles  an  hour,  start  at  the  same  time  from  two  places 
120  miles  apart,  and  move  towards  each  other.  How 
long  will  it  be  before  the  trains  meet  ? 


6. 

1.  Add  2a  —  Bb  +  c,  2b-Sc  +  a,   2c-Ba  +  b, 

and  a-\-b-\-c. 

2.  From  5a3 -7a2b  +  6ab2 -  b3  +  5 

take  3a3  +  4a2b-3  +  8ab2-3b*. 

3.  Multiply  x3  —  2  x2  y  +  2  xy2  —  y3  by  x2  -\-2xy  +  y*. 

4.  Divide  a3  —  b3  —  c3  —  3 abc  by  a—b  —  c. 

5.  Solve   i(;r-4Hi(:r  +  4)  =  TVO*;  +  20). 

6.  Simplify  ^--1---^. 

7.  Solve    #-l-2y  +  3z  =  10 

2x+3y  +  4:z  =  16 
4#  +  4?/  +  5z  =  25 

8.  A  man  bought  3  horses  and  5  cows,  and  gave  the  same 

sum  for  the  3  horses  as  for  the  5  cows.  If  he  had 
bought  4  horses  and  10  cows,  his  outlay  would  have 
been  $  600  more.     Find  what  his  outlav  was. 


EXAMINATION    MANUAL. 


7. 

1.  If  a  =  1,  b  =  3,  c  ==  5,  c?  =  0,  find  the  value  of 

12a3-52  .     2c2  g  +  62  +  c3        g2-^2 

3a2  a  +  62     62-2fo  +  c2      6a-c' 

2.  Simplify  2-3a?-(4-6a?)-{7-(9-2a?)j. 

3.  Multiply  1  -f  2x  +  3a;2  by  1  +  4a;  +  5a:2  +  6a;3. 

4.  Find  the  continued  product  of 

x-\-y,  x  —  y,  and  a;2  +  2 xy -f- y2. 

5.  Divide  a;3  -f-  (a  -|-  ^  +  c)  &  +  («&  +  b°  +  ^)  #  +  °bc  by 

a;  +  a. 

6.  Eeduce  to  the  simplest  form  — ^ — ,      7q  x  ■  a      ,.• 

r  a2  —  ab-\-b2      a3  —  b* 

7.  Solve  the  equation —  =  — r— - 

4  o  o 

8.  A  is  twice  as  old  as  B  and  four  years  older  than  0.    The 

sum  of  the  ages  of  A,  B?  and  0  is  96  years.     Find 
B's  age. 

8. 

1.  Find  the  value  of  a!  +  ?!~C!  +  ^  if  a  =  4tS  =  },c=l. 

a2  —  b2  —  c2  -f  2  be 

2.  What  number  diminished  by  42  becomes  18  ? 

3.  Simplify  \y*-(l-y)]x-\x2  +  (x-y)y\. 

4.  Multiply  px2  +  qx  —  r  by  mx  —  n,  and   bracket  coeffi- 

cients of  the  powers  of  x  in  the  product. 

5.  Divide  1 -far5- 8?/*  + 6a;y  by  l  +  x-2y. 

6.  Solve  3a?~2  +  90a?"-9  =  5s  +  4*. 

7.  Find  the  length  and  breadth  of  a  rectangular  field  if 

the  length  is  140  yards  more  than  the  breadth,  and 
if  it  requires  1000  yards  of  fence  to  enclose  it. 

8.  Point  out  first  the  factors,  secondly  the  terms,  and  thirdly 

the  coefficients,  in  the  expression  5  ax  —  2  ay  +  abz. 


ALGEBKA. 


9. 

1.  If  a  =  1,  b  =  2,  find  the  value  of 

ae  _  3a56  +  6a*b2  -  7a3 b3  -f  6a264-  3a&5  +  5s. 

2.  Divide  £4a+4y45  by  x2a+2^<y  +  2y2b. 

3.  Resolve  into  the  simplest  factors 

ax2~a;  x'-Sla4;  8a3  +  27;  ^  +  7^+10. 

4.  Find  the  L.O.M.  of  a  +  b,  b-c,  c  +  a,  a2-b2,  b2  +  c\ 

c2-a2. 

5.  Find  the  H.C.F.  of 

x2  —  y2 -{■  xz -\-  yz  and  x2  +  y2  +  xz  ~  Vz  —  ^XV- 

6.  Simplify  2(a?-y)-2J3(a;-y)-^-3y-^U. 

7.  Solve  2a(3a-a7)-3^(26-a)  =  a5. 

8.  The  sum  of  .two  numbers  is  70,  and  their  difference  is 

23  ;  find  the  numbers. 

10 

1.  If  a  =  25,  5  =  9,  c  =  4,  d  =  1,  find  the  value  of 

3  Va  +  2  VU  -  V9  c  +V16  d. 

2.  From  7a;3  -  6x2  +  2a?  -  1  take  ±x3 -3x2  -  x  +  2. 

3.  Multiply  a2n  +  ana;w  +  a;2n  by  2an  -  2xn. 

4.  Divide  xK  —  9  a;2  —  6xy  —  y2  by  x2  +  y  +  3  a?. 

5.  Resolve  into  the  simplest  factors 

x't-y16;  8x«  +  -3)  lQxn-yi;  x2  +  6x-7. 

6.  Find  the  H.C.F.  of 

6a2  +  7aa?-3a?  and  6  a2  +  11  as; +  3  a;2. 

7.  solve  2x  +  a      (*  ~  h)  ~  SaX  +  (a  ~  5y 

b  a  ab 

8.  A  is  now  twice  as  old  as  B,  but  10  years  ago  he  was  three 

times  as  old.     Find  the  present  ages  of  A  and  B. 


EXAMINATION   MANUAL. 


11. 

1.  If  a  =  0,  b  =  2,  c  =  4,  rf=  6,  find  the  value  of 

3</2b2-a  +  2$V+j  +  1  -  ^/2(b  +  c)2~(b  +  d)2. 

2.  From-fa:4-f-£a;2  +  3a:-lf  take  %x*-%x-2\. 

3.  Divide  l-6x5+5Vby  \-2x-\-x2. 

4.  Simplify  (a;  +  y)2  -  2*  (3a;  +  2y)  -  (y  -  x)  (-  x  +  y). 

5.  Resolve  into  the  simplest  factors 

2^-3#-5  and  a2  -  b2  +  c*  -  2ac. 

6.  Find  the  H.O.F.  of 

yK  -\-m2yl  -\-  ra4  and  y4  +  my3  —  mzy  —  m4. 

7.  Solve  5_.2£zil!_£l±3  =  o. 

2  3  4 

8.  A  person  bought  16  yards  of  cloth,  and  if  he  had  bought 

one  yard  less  for  the  same  money,  each  yard  would 
have  cost  25  cents  more.  What  did  the  cloth  cost 
per  yard  ? 

12. 

1.  Find  the  H.C.F.  of 

12a2+lSab  +  3b2  and  6 a2  +  23 ab -f  W. 

2.  Simplify  2(x-y)-2{s(x-y)-(2x-Sy-^\  1 . 

3.  Find  the  square  root  of 

9  a4  -  12  a*b  +  34  a2b2  -  20  ab3  -f  25  64. 

4.  A  is  twice  as  old  as  B  ;  22  years  ago  he  was  four  times 

as  old ;  what  is  A's  age  ? 

5.  Multiply  x2  +  ^  -  y2  by  x2  -  ^  +  ^ . 

6.  Solve    ar  +  ii^  =  ^^- 

7.  What  fraction  is  that  to  the  numerator  of  which  if  7  be 

added,  its  value  is  I ;  but  if  7  be  taken  from  the 
denominator,  its  value  is  f  ? 

8.  Solve   #  +  2?/  =  23,   3#  +  4z  =  57,   5y  +  6z  =  94. 


ALGEBRA. 


13. 

1.  Add  .5a;3-  3a;  +  2y,  -x*  +  2x  -y,  and  7a;3  -4a; -j-  3y. 

2.  From2a;  +  lla  +  10&-5e-23  take2c-10+5a-36. 

3.  Multiply  9x2  +  Sx7/  +  f-6x  +  27/  +  4:hy  Zx  —  y  —  2. 

4.  Divide  a;3  —  5a;2  —  x  +  14  by  a;2  —  3a;—  7. 

5.  Solve  K^+l)  +  K^+2)-16  +  i(a;  +  3)  =  0. 

6.  Resolve  into  factors  2xy  —  x2—y2-\-  z2. 

7.  Simplify  ^ +  —£-  +  — ^ — 

y     x-\-y     x2-\-xy 

8.  Solve3a;-22/=:5 

4a;-3y+2z=ll 
x—  2y  —  5z  =  —  7 

14. 

1.  Find  the  square  root  of  a;4  +  2x%  —  x  +  -J-. 

2.  Find  the  H.C.F.ofa;4+3a;3-6a;-4anda;4-3a;3  +  6a;-4. 

3.  Resolve  into  factors  x2  -f-  20  a;  +  91. 

4.  Divide  86  into  two  parts  such  that  their  difference  shall 

be  18. 

5.  Simplify  «=*  +  _^L.  _ ^!+^. 

o         a  —  b      a2b  —  b3 

6.  Divide  1 —  by  1  +  — S- — 

a;2  +  a2    J         a;2-a2 


7.    Solve  ar  +  y  +  z  =  90 

2a;  +  40  =  3y  +  20 

2* +  40  .—  4* +  10 


8  EXAMINATION   MANUAL. 


15. 

1.  Multiply  a3  -  2a2b  +  3ab2  +  U3  by  a2  -  2ab  -3b\ 

2.  Divide  x4  +  (2b2- a2) x2  -f  b*  by  x2  +  ax  +  62. 

3.  Find  the  square  root  of  —  +  8  +  — 

62  a2 

4.  Solve   (3a;-5)(6^-7)-2(3^-5)2  =  7. 

5.  Solve   3x  —  y  +  z  =  l7 

5x  +  3y~2z  =  l0 
7^  +  4y  —  5z  =   3 

6.  Kesolve  into  factors  x2  —  y2  -f  z2  —  a2  —  2xz  -f  2  ay. 

7.  Simplify 

1  2,1 


+ 


(2-ra)(3-m)      (m-l)(m-3)      (m-l)(m-2) 

16* 

1.  A  body  moves  a  feet  toward  the  north  and  then  b  feet 

towards  the  south.  How  many  feet  is  the  body,  at 
the  end  of  the  motion,  from  the  starting  point  ?  On 
which  side  of  the  starting  point  is  the  body  (i.)  if 
a  >  6,  (ii.)  if  a  <  b  ? 

2.  Reduce  to  the  simplest  form 

l-{l-(l-4:x)\+{2x-(S-5x)\-\2-(-4:+5x)\. 

3.  Divide  a2  +  ab  +  2ac-2b2  +  7bc-3c2  by  a-  b  +  Sc. 

4.  Resolve  into  elementary  factors  a6  —  9  a4b6. 

5.  Find  by  inspection  the  H.C.F.  of 

4  O3  +  a3)  and  6  (x2  -  2ax  -  3a2). 

6.  Simplify  £±y___2g_  +  s*y-< 

y  %-\-y      x2y  —  y3 

7.  Solve   T\(2x~S)-^(3x-2)  =  }(4:x-S)-S^. 

8.  A  and  B  have  together  $8,  A  and  C  have  $10,  B  and  C 

have  $12.     What  have  they  each  ? 


ALGEBRA.  9 


17. 

1.  Resolve  into  four  factors    (a2  —  b2~  c2)2  —  4  b2  &. 

2.  Find  the  H.C.F.  of  4:X*-xy2  and  4^3  -  10x2y  +  4:xy2. 

3.  Simplify    Sa-(a~b-c)-2\a  +  c-2(b-c)\. 

4.  Divide   x8  +  xUf +  y*  hy  x*  + x2y2  +  y\ 

5.  Solve   2^-^=7  +  ^2*   - 

4  5 


4y-f 


a?-2  =  26J_22M1l 


3  2 

6.  Divide  91  into  two  such  parts  that  the  quotient  of  the 

greater  part  divided  by  the  difference  between  the 
parts  may  be  7. 

7.  Extract  the  square  root  of  16  x*  —  32  #3  -f-  24  #2  —  8x  -f- 1. 


18. 

1.  From  ax2  -f  bxy  -f  cy2 

take  (b  —  c)x2-\-(c  —  a)  xy-\-(a  —  b)  y*. 

2.  Multiply  x2  —  (a  —  b)  x  —  ab  by  x2  +  (a  —  5)  x  —  a&. 

3.  Resolve  into  the  simplest  factors 

32#5  +  2/5;    hx2  +  %xy-2ly2;    6a2x2-7ax-S. 

4.  Simplify  3a-52a-(3a-Z02S+3aJ25-3a--^-l. 

I  3a  J 

5.  Find  the  L.C.M.  of  3(1 -a;2),    6(1 -a;)2,    5(a:+a;2). 

6.  Find  the  H.C.F.  of  7a;2- 12*;  + 5  and  2x3+x2^8x-\-5. 

7.  Solve   2>x-^{x-n)  =  9~\(px-l). 

8.  If  A  can  do  a  piece  of  work  in  8  days,  and  A  and  B 

together  can  do  it  in  6  days,  how  long  would  B 
take  to  do  it  alone? 


10  EXAMINATION   MANUAL. 

19. 

1.  From  axm+n  +  (a  -f  b)  xm  +  2  bxm~n 

take  bxm  +  (a  +  b)  xm~n  —  ax™-271. 

2.  Find  the  coefficient  of  x*  in  the  product  of 

a*  -  bx2  -  3  abx  +  a2  and  2s;2  -  3  ax  -  b2. 

3.  Reduce  to  lowest  terms  a'2+  h*~  c*+  2ab. 

a2-b2-c2+2bc 

4.  Simplify  ^+     *         <*±£ 

5.  Find  the  H.O.F.  of 

7a2-23a£  +  652and5a3-18a2£  +  llaZ>2-663. 

6.  Reduce  to  a  single  fraction  a  —  b  -) 

7.  Solve    _?£---- 3  =  —^~ 


4a 


26  2a-6 

8.    A  square  floor  would  contain  17  square  yards  more  if 
each  side  were  1  yard  longer.     Find  its  area. 


20. 

1.  Divide  2aSn-  6a2nbn  +  6anb2n~  2b3n  by  an-bn. 

2.  Find  the  H.O.F.  of  20x5+xs-x  and  25^+5.^- x2 -x. 

3.  Simplify  2o^+8a«  x  4tts»-3a'a»  x     a2 


4:x2-Sax         a2x2-ct         2*2+3a# 
4.   Resolve  into  factors  256  x8  —  1  and  x2 — 7  x  — 18. 


5.    Simplify 


f  *    |    y  V  (  x        y  \ 

\x  +  y      x-yj      \x-y      x  +  y) 


6.  Solve   (a  +  x)(b  +  x)~a(b  +  6)  =  ?^  +  x2. 

b 

7.  A  laborer  was  engaged  for  36  days  on  condition  that  for 

every  day  he  worked  he  was  to  receive  $3,  and 
for  every  day  he  was  idle  to  forfeit  $2.  At  the  end 
of  the  time  he  received  $68.  How  many  days  did 
he  work  ? 


ALGEBRA.  11 


21. 

Resolve  into  their  simplest  factors 

a3  -  a2  x  -  ax2  +  xz ;    x*  +  3  x2  y  -  4  :n/2  -  12  y\ 
a;-f-l        a  — 1  1  — 3a; 


2.    Simplify 


2a?-l      2a; +1      a;(l  — 2ar) 


Simplify   ../„    ,    n+„/.,       ,2x  + 


«0  +  6)      a(a2-b2)      b(a-b) 
4    Solve   8a-  +  5      7*-3_  16*;+ 15      2£ 
14         6*  +  2  28  7 

5.  Find  the  L.C.M.  of 

Sx^-Ux+6,  4:x2  +  ±x-3,  and  ±x2  +  2x-6. 

6.  Add  l-{2-(3-a?)j,    3#— (4-5*),   4-r-(-5  +  6ar). 

7.  Twenty-four  persons  subscribed  the  cost  of  a  new  boat, 

but  four  of  the  subscribers  proving  defaulters,  each 
of  the  others  had  to  pay  $  2  more  than  his  due  share. 
Find  the  cost  of  the  boat. 

22. 

1.  Resolve  into  their  simplest  factors 

aV-fiy4;   27bxi-bixys;   20ax2+l8a2x-18a\ 

2.  Simplify  V        XJ,        ^      XJ- 

3.  Find  the  L.C.M.  of  x2-±y2  and  x2  +  xy-6y2. 

4.  Solve    14(7ar+5)-[84a;-(9a?-21)]  =  210. 

5.  Solve  **±±-  4^  +  2  =  2£lL5 

3  Sx-9  3 

6.  A  and  B  can  do  a  piece  of  work  in  8  days,  working  to- 

gether ;  A  working  alone  can  do  it  in  12  days.  In 
how  many  days  can  B  do  it,  working  alone  ? 

7.  State  the  distinction  between  an  algebraic  identity  and 

an  algebraic  equation,  and  give  an  example  of  each. 


12  EXAMINATION    MANUAL. 

23. 

1.  Ifa  =  0,  b  =  2,  c  =  -S,  d  =  4,  find  the  value  of 

4:(ad-bc)-{(a-b)-(c-d)\\ 

2.  Add  $a  +  $b  —  \e  + Jrf-andf  «  — f  &  +  c  —  fi 

3.  Divide  a;6  -  2  a3  ^3  +  a6  by  x1  -  2  a^  +  a\ 

X*  ~\~  Q? 

4.  Eeduce  to  the  lowest  terms 


x2jr2ax-\-  a2 

5.  Solve   4a?  +  9y  =  121  # 

6^-32/=    7j" 

6.  Extract  the  square  root  of  x4,  —  2xz  -| — —  +  —  • 

7.  A  person  walking  at  the  rate  of  4f  miles  an  hour  starts 

li  hours  after  another  person  who  walks  only  4 
miles  an  hour.  When  and  where  will  he  overtake 
him? 


24. 

1.  If  a  =  1,  b  =  3,  c  =  5,  d  =  7,  find  the  value  of 

a- 25-  {3c-^~[3a-(56-c-8c?)]-25}. 

2.  Simplify  (*-8)  — (8-*)(l.-a?)-(a?-8)(5-2a?). 

3.  From  (&+6)2-2&r  take  \(a-\-b)(a-x)-(a—b){b—x)\. 

4.  Find  the  H.C.F.  of 

I2xy(x,y-2xiy*  +  xtf)1  20(x*-2xhyi  +  x1y«),  and 
&y@-xf). 

5.  Solve   E±^_g-y-2=i 

12  5 

9y_5^  =  10.     • 

6.  Find  a  number  which  is  as  much  greater  than  63  as  its 

half  is  less  than  93. 


ALGEBRA. 


13 


25. 

1.  Resolve  into  the  simplest  factors  x12  —  y12. 

2.  Simplify  a2  -  (b2  -  c2)  -\b2-  (c2  -  a2)  \  +  \c2  -  (b2  -a2)}. 

3.  Find  the  H.C.F.  of  8aV  +  12aV  and  12a;5  +  18atf*. 

4.  Divide  a2  +  ah  +  ac  +  bc  ,     a_+i 

a2  —  ae  +  ac?  —  cd        a-\-d 

5.  Solve   2ax  +  b  =  3cx-}-4:a. 


Solve    -  +  -  =   7 
a;     y 

The  sum  of  the  two  digits  of  a  certain  number  is  six 
times  their  difference,  and  the  number  itself  exceeds 
six  times  their  sum  by  3.     Find  the  number. 


26. 


1.   Solve 


2^3 


2  +  *=4 
2^8 

2 X   1 

16     l2~6 

2.  A  person  swimming  in  a  stream  which  runs  \\  miles  an 

hour  finds  that  it  takes  him  four  times  as  long  to 
swim  a  mile  up  the  stream  as  it  does  to  swim  the 
same  distance  down.  Find  his  rate  of  swimming  in 
still  water. 

3.  Find  the  H.C.F.  of  3z4-4a;3+l  and  4^-5^-.^+^+ 1. 


4.    Solve 


x  —  7  .  x x-\-  7 

2     ^9"     3 


5.  Divide  a2  b  +  (a  —  b)2  x 

6.  Simplify  £±£— 


iCbsr- 

x 


x3  by  b  -J-  x. 


y    z  +  y 


14  EXAMINATION   MANUAL. 

27. 

1.  Subtract 

(a+y>3-(2  a2-f)x2~2  arf  from  aty-(2  x2-a)y2+  ax\ 
and  arrange  your  answer  in  descending  powers  of  y. 

2.  Divide  a2 -  ±ab  +  U2  by  a2~2ab. 

a2      b2  a +  26 

3.  Solve =  a  —  b. 

x       x 

4.  Simplify  X1±3L  +  ±ZlL 

x  —  y       (x-y)1 

5.  Find  two    consecutive    numbers,  the   difference  of  the 

squares  of  which  is  equal  to  51. 

6.  Two  trains  pass  a  station  at  an  interval  of  4  hours,  trav- 

elling respectively  at  the  rates  of  11 J  and  17  J  miles 
an  hour.  How  far  will  the  slower  train  have  run 
before  it  is  overtaken  by  the  other  ? 

7.  Solve   |*  +  $y  =  16;    \x-\y  =  2. 


28. 

1.  Divide  x3  by  1  —  x2  to  three  terms.     What  is  the  differ- 

ence between  the  answer  thus  obtained  and  the  true 
answer  ? 

2.  Arrange  according  to  powers  of  x 

3(a2-2x)(a-x*)  +  2x(a-6x2). 

3.  Find  the  H.C.F.  of 

6a3  -  6a2y  +  2ay2 -  2tf  and  12a2-  Way  +  3y2. 

4.  How  do  you  find  the  L.C.M.  of  two  expressions  when 

their  factors  cannot  be  determined  by  inspection  ? 

r-     a.      ,-.-       a?  +  a3    sx  xz  4:a2x2-\~2ax3 

5.  Simplify   - — ! X ■ — -  -= — -J — - — 

Aax  —  xl      xz  —  ax-\-al  4a -r 

6.  A,  B,  and  0  can  reap  a  field  in  30  hours.     A  can  do 

half  as  much  again  as  B,  and  B  two-thirds  as  much 
again  as  0.  How  many  hours  would  each  require 
to  do  the  work  alone  ? 

7.  Solve   x-}(y-2)  =  5;     iy  -  $(s  +  10)  =  3. 


ALGEBRA.  15 


29. 

1.    Divide  aL±I  +  a-^  by  2±1  _£=£. 
<z  — a;      a-\-x         a  —  x      a-\-x 

1      ,       x 


2.    Simplify   l~\X      l      X* 


1-x2      l  +  x 

3.  Solve   5^  =  43-7y;     ll#  +  9y  =  69. 

4.  Find  the  H.O.F.  of  %o*-\0a*+\ba?-&c  and  2x6-Ax4+2x2. 

5.  Find  all  the  expressions  whose  L.C.M.  is  x3  —  4xy2 

6.  A  number  consists  of  two  digits.     The  sum  of  the  digits 

is  7,  and  if  27  be  subtracted  from  the  number  the  or- 
der of  the  digits  will  be  reversed.    Find  the  number. 

3  3 

7.  Find  the  value  of j ■    when  x  —  £. 

l-\-x      1—x 


30. 

1.  A  is  x  years  old ;  B  is  20  years  younger.     Express  in 

algebraic  language  the  age  of  B ;  the  sum  of  their 
ages ;  the  age  of  A  five  years  ago ;  the  age  of  B 
five  years  hence. 

2.  Simplify  (2a-b)2+2b(a+b)--Za2-(a-b)2+(a+b)(a-b); 

3.  Multiply  a--  by -+-. 

4l    Simplify  ct(x2  +  c2)-2acx^ 
F    y  b(x2  +  c2)-2bcx 

5.  Find  the  H.C.F.  of 

3a3-7a26-33a&-7a62+ll£2-{-363  and  3a2-l0ab+3b2 .. 

6.  Solve  5x-\-2y—  z  =  2z  —  x  —  y  —  y  +  z  —  2x  —  6. 

7.  If  a  father  was  four  times  as  old  as  his  son  seven  years 

ago,  and  will  be  twice  as  old  as  his  son  in  seven 
years  more,  what  is  the  present  age  of  each  ? 


16  EXAMINATION   MANUAL. 

31. 

1.  What  is  meant  by  a  term,  an  index,  a  factor?     Give 

examples. 

2.  Solve  ™  +  ^  =  «; 
x      y 

3.  Solve  2x-  4y  + 
7x+  3y- 
9o;  +  10y-llz  = 

4.  Find  the  H.C.F.  of 

2a;2  +  3xy  +  6a;  +  9y  and  3^  -  2a;y  -f  9a:  —  6y. 

5.  Simplify    2   **-/     a^E±£ 

a;2—  2a;y-f-y2       x6  —  ^ 

6.  Find  a  fraction  such  that  if  5  be  added  to  the  numerator 

the  fraction  equals  1,  but  if  3  be  taken  from  the  de- 
nominator the  fraction  equals  i. 

7.  Arrange  according  to  descending  powers  of  x 

x2  —  ax  —  c2x2  —  bx-\-  bx3  —  ex2  -f-  a2xz  —  x2  —  ex. 

32. 

1.  What  is  a  common  multiple  of  two  or  more  quantities? 

What  is  meant  by  their  least  common  multiple  ? 

2.  Find  the  H.C.F.  of 

4a;4+22a;3-26a;2-198a;-90  and  4o;3-14o;2-92a;-42. 

rnn 


(m+nf 


a2c 


3.  Bwn%-H_+/_»_y. 

m  +  n      \m  +  nj 

4,  Solve   (a  +  x)(b  +  x)-a(b  +  c)=~+xr. 

b 

5l    Solve   ^  +  |  =  5;      §-4y==i 

6.  Solve   a-  +  y  =  16;     2z-2a;  =  8;     2y  +  3z  =  51. 

7.  At  an  election  there  were  two  candidates,  and  1296  per- 

sons voted ;  the  successful  candidate  had  a  majority 
of  120.     How  many  votes  were  cast  for  each  ? 

8.  What  is  the  rule  for  transposing  terms  in  equations? 

What  is  the  reason  for  the  rule  ? 


ALGEBRA.  17 


33. 


a-\-x  .  a—x 


1.  Solve    ^  +  ^  =  a;__^_. 

a  —  b      a-\-b  a1  —  b2 

2.  Solve   x  +  y  =  a ;     bx  —  cy  =  d. 

3.  Solve   5*-%-f2s=*19;  4<*f5y-3*=81;  3a+7y-4z=31. 

4.  Simplify  a  +  h     a~b       *ab 


a  —  b      a-j-b      b2—a2 

5.  Eeduce  to  lowest  terms    21s*+14syt 

18a^+12ay 

6.  Divide  x«-  2a3 x3  +  a6  by  x2  -  2 ax  +  a2. 

7.  Simplify  2\a{a~b)-{a-{-b)2\{a2-2b2\. 

8.  The  difference  between  two  numbers  is  one-fourth  of 

their  sum,  and  the  half  of  the  greater  diminished  by 
7  is  equal  to  one-fifth  of  the  less  increased  by  2Jr . 
Find  the  numbers. 

34. 

1.  Define  factor,  common  factor,  highest  common  factor. 

2.  Find  the  H.C.F.  and  the  L.C.M.  of 

(2  a2  bx  +  2  aba*)*  and  (6  a2  b2  -  6  J2  x2)3. 

3.  Multiply  a6+  a5-  a  -  1  by  1  -  a  +  a2-  a3  +  a\ 

4.  Simplify - 


1-M4 


1  +  *  + 


1+37 

5.  Solve   "«  +JL-S*+* 

era;      or# 

6.  Solve   x  +  y  =  5  (a?  —  y)  =  10. 

7.  A,  B,  C,  and  D  have  $290  between  them.     A  has  twice 

as  much  as  C,  and  B  has  three  times  as  much  as  D. 
Also,  C  and  D  together  have  $50  less  than  A.  Find 
how  much  each  has. 

8.  Expand  (2x2yzJ-,    (-7a86V)s;    (-3a5V)*. 


18  EXAMINATION   MANUAL. 


35. 

1,    Simplify  tx    a  +  b f      h  +  e + 


(b-c){a-c)      (a-b)(c-a)      (b-a)(b-c) 

2.  Simplify  q'+3q'6*3qy  +  y      a2-2ab  +  b2, 

P    *  a3-3a26  +  3a&2-63      a2  +  2a6  +  62 

3.  Solve  i*+iy=12-iz;  £y+t*  =  8+£z;  ^+^  =  10. 

4.  Find  the  L.C.M.  of  9a2-81,  12a2-36a,  and  16a2+48a. 

5.  Extract  the  square  root  of  xi-8x*-2e>x2+W8x+4Al. 

6.  Two  persons  start  at  noon  from  towns  60  miles  apart. 

One  walks  at  the  rate  of  4  miles  an  hour,  but  stops 
2 £  hours  on  the  way.  The  other  walks  at  the  rate 
of  3  miles'  an  hour,  without  stopping.  When  and 
where  will  they  meet? 

7.  How  many  different  equations  must  be  given  when  the 

values  of  two  or  more  unknown  quantities  are  re- 
quired ?     In  what  sense  must  the  equations  differ  ? 


36. 


3/— 


3  I- 

1.  Find  the  value  of  x-(Vx±l+2)-x    ^  x  when  x  =  S. 

Vx— 4 

2.  Divide  a2  x*—  c2  y*—  bx2  y(2ax—  by)  by  (ax  —  by)  x  —  cy1. 

3.  Find  the  H.C.F.'of 

Sax2+2mj2-Sbx2-2by2  and  3azs-2ay2-3bx2+2by2. 

4.  Find  the  L.C.M.  of  a2+x2,  a2-x2,  a?+x3,  and  az-x\ 

5.  g^phfy   —  x  a2__2x_a{a_2)  X  (g,+  ^(g+g)- 

6.  A  is  5  years  more  than  twice  as  old  as  B,  and  is  also  25 

years  older  than  B.     Find  their  present  ages. 

7.  "What  are  simultaneous  equations  ?     Give  an  example. 

8.  Expand  («  +  |J. 


ALGEBRA.  19 

37. 

.,     q.      yn     be | _ae ab 

'      impiy  (a-b)(a-cy  (b-a)(b-c)      (b-e)(c-a) 

'a      b\(a  +  b      a  +  b' 

J)      aj\    a 
2.    Simplify   7^       b,n'    ± 


b2      a2 J  \a      b 

3.  solve  i+M-g;  M+5-&  hhs=^- 

x^y^z      20     y^z^x     60     z^ x^ y     30 
"4.    If  a  rectangular  court  had  its  length  diminished  by  7 
feet,  and  its  width  increased  by  5  feet,  it  would  be- 
come an  exact  square,  enclosing  the  same  area.    Find 
the  dimensions  of  the  court. 

5.    Extract  the  square  root  of  x^  —  x3  -j-  — +  -• 


6.   Expand  g-fj 


2  '  4 


38. 


1.  Find  the  product  of  —  —  — --  and  — ■—  -\ f-  -f-  -£r« 

a        o  a1        ab         bl 

2.  Find  the  H.C.F.  of  x*+e>x2+llx+6  and  ar,-|-9rc2+27a?+27. 

q    «•      Vo     3:z2  +  l      hx-2  ,  2x  +  b 

3.  Slmphfy  _____  +  _^. 

4.  Extract  the  square  root  of  x*  +  — -  -j — f-  -  -f-  -• 

3  9        3      4 

£2 

5.  Solve   ax x  ~  a  -f-  5. 

« 

6.  A  and  B  have  each  an  annual  income  of  $4000.     A 

spends 'every  year  $400  more  than  B.     At  the  end 
of  4   years  their  joint  savings  amount  to  $4000. 
What  does  each  spend  annually  ? 
/  3V 

7.  Expand  [2a  — 


8.    Explain  how  to  clear  an  equation  of  fractions,  and  the 
reason  for  the  method. 


20  EXAMINATION    MANUAL. 


1.    Simplify 


39. 

1  ,  1 


a(a—b)  (a—c)      b(a—b)  (c—b)      c(c—a)  (b—c) 
SI       #  — 3      x—  6_*  +  12      x  —  6  .  x 
0VG       7  6-    ~     12  3     ^8 


9h 


3.  Solve   7  +  |  =  8;      -  +  ^  =  6. 

4      2  2     6 

4.  Extract  the  square  root  of  — 2  + 

5.  Simplify  1- {l-(l-4x)\+{2z-@-5x) }-{2-(-4+5s){/ 

6.  Find  the  L.O.M.  of  a'-l,    (aJ  -  5)  <?,  and  (ac  +  c)  b. 

7.  A  and  B  can  mow  225  square  rods  in  a  day ;  B  and  C 

can  mow  223  square  rods  in  a  day ;  A  and  C  can 
mow  230  square  rods  in  a  day.  How  many  square 
rods  can  each  mow  by  himself? 


40. 

1.  Simplify  M\a(a-b)(Za-b)-(a-by\. 

2.  Find  the  H.C.F.  of 

12  a2  +  13  ab  +  3  b2  and  6a2  +  23  ab  +  lb2. 

3.  Find  the  L.C.M.  of 

4(1  -x)\  8(1-*),  8(1  +  *),  4(1 +  *2). 

4.  Find  the  cube  root  of 

l-6*  +  21*2-44*3+63*4-54*5+27*6. 

6.  Solve    |-1W|  +  1  =  |;    x  +  y  +  z^-ll. 

7.  At  what  time  between  two  and  three  o'clock  are  the 

hour  and  minute  hands  of  a  clock  in  a  straight  line 
and  pointing  in  opposite  directions  ? 


ALGEBRA.  21 


41. 

1.  Find  the  square  root  of  1  —  xy  —  *£■  x2y2  -f  2x*y*  -f  4or4y4. 

2.  Simplify  K2  -  i°0  -  a  _  fr(6a  +  4)  +  ia  +  2 

r    *  2-i(2  +  2o)  l  +  2a 

3.  Solve   ^  +  y  +  2  =  6;     #  -f-  y  —  z  =  4 ;     a;  —  y  -}-  z  =  2. 

4.  Multiply  f  -|  byf  -|  +  1 

5.  A  boat  is  rowed  1  mile  in  5-^  minutes ;  another  is  rowed 

'  1  mile  in  5}  minutes.  If  they  start  at  the  same  time 
from  opposite  ends  of  a  4-mile  course,  at  what  dis- 
tance from  each  starting-point  will  they  me*et  ? 


6. 


Expand  (f  +  f J. 


7.   Solve  o2/  +  ®  +  ^x  ~  aN  =  24. 


1.    Simplify 


42. 
3 ,      x         x—3 


2.    Find  the  H.C.F.  of  Ss?-3a?y+xyi-yi  and  itf-xy-Sy2. 


,  a&2      &3 


3.  Find  the  cube  root  of  a3  —  a2  b  A 

3       27 

4.  Solve   (2  +  x)(8  +  x)-22  =  %  +  x\ 

5.  Solve  the  simultaneous  equations 

3*  =  2<j/  +  2)--|;  2y=3(*~*)-Hh  2=^+3/ 


-t 


6.    Find  the  value  of  2  (1  +  Va)  -  ^""^ 


a 


3  Va  +  vV 

when  a?  =  lo  a. 

7.  36  octavo  volumes  exactly  fill  a  box  whose  length  and. 
breadth  are  1-J-  feet  and  11-J-  inches,  respectively. 
The  dimensions  of  each  volume  are  9  inches  by  5| 
inches  by  1|  inches.     Find  the  depth  of  the  box. 


22  EXAMINATION    MANUAL. 

43. 

1.  Find  the  L.C.M.  of  9a*+5Bx*-9x-rl8  and  fcM-llar+3(X 

2.  Find  the  square  root  of  x*  —  4  xz  -f  re8  +  6  #  + f . 

3.  Solve  Sx+4y-6z==Z2]  4^-5y+32=18;  5;r-3y-42=2. 

4.  Divide  5^+^-^7«V+^+^by2^-^-^ 

5.  Simplify 


6  —x 

x  — 1,3/  — 1,2- 1 

6.  Simplify  §32 ~      "T"      ^~ 

yz-\-xz  —  xy  1      1      1 

#      y      z 

7.  Eeduce  to  lowest  terms  — — 

6;r  —  x—1 

44. 

1.    Simplify  — JL -  + 


(2a;  -l)2      (2a7+l)(l-2s) 


o    tv   •  i     4a2  — 4a#  , 
2.    Divide  -— —   by 


(a  +  xf  4aa:4-4a;2 

o      \iji       *        a2     b2  .  c2     b2     c2  ,  a2     c2     a2  ,  b2 

3.  Add  together \-—, — . 

8  2      3      4      2      3      4      2      3      4 

4.  Find  the  L.C.M.  of 

xz  -f  x2y  +  a;?/2  +  y3  and  #3  —  tf2?/  +  a:?/2  —  y3. 

5.  Extract  the  square  root  of  a2+4  52+c2+26zc+4  a6-f-4  be. 

6.  Solve   -  = 

x — b      x—a 

7.  Divide  the  number  90  into  four  parts  so  that  the  first 

part  increased  by  2,  the  second  diminished  by  2,  the 
third  multiplied  by  2,  and  the  fourth  divided  by  2, 
may  be  all  equal. 

8.  What  powers  of  a  quantity  are  the  same  whether  the 

quantity  be  positive  or  negative  ? 


ALGEBRA.  23 


45. 

1.  Simplify I 4- 1 

r-    Jl  +  3a  +  2a2  '   l-a-2a2 

2.  Multiply  1  —  x  —  x2  by  x2  —  a?  —  1. 

3.  Find  the  H.C.F.  of 

xH  +  bx2-{-10x-{-8  and  x5  +  2xi-x-2. 

4.  Find  the  L.C.M.  of 

l-x2,  \-2x  +  x2,   l-x  +  x2-x\  l-x-tf  +  x3. 

5.  Reduce  to  lowest  terms  "x  ~XV~V  . 

y2  -\-xy-2x2 

6.  Expand   (<*-£)' 

7.  Solve  i  +  l_i  =  6;    1  +  1-1  =  4-    1+1-1  =  2 

y     z      x  z     x     y         '    x     y      z 

46. 

1.    Reduce  to  a  simple  fraction  x 


2.  Reduce  to  its  lowest  terms  ^     %x*  +  7x — §1 

2a*  +  19  a* +  35 

3.  Find  the  square  root  of  J~\      ,   , 2£x_lL 

16#2y2 

4.  Solve    3*-a  =  ^±^-^-^. 

3  a 

5.  The  united  ages  of  a  father  and  son  make  75  years. 

The  father  was  25  years  old  when  the  son  was  born. 
"What  is  the  age  of  each  ? 

6.  Solve   bx  -Zy  -28  =  0) 

3a;-  2y-  13  =  0  /' 

7.  Subtract  ax  from  2«* +  «'*»  + 2*        , 

2-\-ax-2x2 


24  EXAMINATION   MANUAL. 

47. 

1.  Multiply  a*+a5-a-l  by  1-  a  +  a2  -  a3+a*. 

2.  Find  the  H.O.F.  of 

4^4+22.t3-26^2-198^-90  and  4z3-14:r2-92tf-42. 

3.  Find  the  L.O.M.  of 

x2-l,     x3~l,     x*+l,     (x+1)2,     (x-1)2. 

4.  Solve      x  +  3y  +  2z  =  11 

2x+    y+3z  =  14 
Bx  +  2y  +    z  =  ll 

5.  What  sum  is  that  which  is  as  much  greater  than  $2  as 

its  half  is  less  than  $  3  ? 

6.  A  number  consisting  of  two  digits  when  divided  by  the 

sum  of  the  digits  gives  7  for  the  quotient ;  and  the 
number  formed  by  interchanging  the  digits,  when 
divided  by  6,  gives  4  for  the  quotient.  Find  the 
number. 

7.  What  powers  of  a  quantity  are  always  positive  ?    What 

powers  of  a  negative  quantity  are  always  negative  ? 


1.   Simplify 


48. 


x+1      x—1      1 

Simplify 


/      x2-y2        .  ff2+3aA  y ,(x5-9xs  .  x^-Sx2^ 
\x2—2xy-\-yi  '    x-3  J      \  x+y  x—y  J  ' 


S0lVG    2^i-l  +  3^+L+l" 

2x2—3x- 
Reduce  to  lowest  terms 

Simplify 


5x*-7x-e> 
^  +  5a7  +  4    .  a?+2x+\ 
#2  +  7z  +  12  '  #2  +  3#  +  2' 

6.   simplify  72  +  f  +  m^+m;     What  wo»ld  the  re" 
bm2  +  on2  +  m2  n2  +  nr 

suit  be  if  a  =  —  m2  ? 


ALGEBRA.  25 


49. 

t  •  «  -,        a  +  h      i    fa  —  b\  be        -      a 

L   Solve   —^--|c   —— ]  =  -f— — . 

26  \  ox  J      {a-\-o)x      a  +  b 

2.  Solve   «#  +  £?/  =  <?;     a'#  +  £'y  =  c'. 

3.  Solve   #+2/+z  =  5;    3#-5y+7z=75;    9#-llz+10  =  0. 

*   a  i        1,1  1,1  1,1 

a;      y  y      z  x      z 

5.    Solve 

3  +  5+7  '     4+6  +  3      ''2      5  +  40         *' 


50. 

1.  A  person  walking  at  the  rate  of  4f  miles  per  hour  starts 

H  hours  after  another  person  who  walks  only  4 
miles  an  hour.  When  and  where  will  he  overtake 
him? 

2.  A  number  consists  of  three  digits,  the  right-hand  digit 

being  0.  If  the  left-hand  and  the  middle  digits  be 
interchanged,  the  number  will  be  diminished  by  180. 
If  the  left-hand  digit  be  halved,  and  the  middle  and 
right-hand  digits  be  interchanged,  the  number  will 
be  diminished  by  454.     Find  the  number. 

3.  A  had  twice  as  much  money  as  B ;  but  when  B  had  re- 

ceived $5  from  A,  he  had  three  times  as  much  as  A. 
How  much  had  they  each  at  first  ? 

4.  A  tank  of  water  may  be  filled  by  three  pipes.     By  the 

first  pipe  it  can  be  filled  in  4  hours,  by  the  second 
in  10  hours,  by  the  third  in  15  hours.  In  what  time 
will  it  be  filled  if  all  three  pipes  are  opened  ? 

What  will  be  the  solution  of  this  question  if  the 
general  symbols  m,  n,  p  are  used  in  place  of  the 
numbers  4,  10,  15,  respectively? 

5.  What  are  simultaneous  equations  ?     Give  an  example. 


26  EXAMINATION   MANUAL. 


51. 

1.  Bracket  like  powers  of  x  in  the  expression 

ax3  —  bx2  —  ex  —  bx3  +  ex2  —  dx  +  ex3  —  dx2  —  ex. 

2.  Divide   a3-  63  +  c3  +  3a£c  by  a-  b  +  <?. 

3.  Simplify  g-  (g=Q*  +  ff -y?f 

4.  Divide  150  into  two  parts  such  that  if  one  is  divided  by 

23,  and  the  other  by  27,  the  sum  of  the  quotients 
is  6. 

5l   Solve   gg-15**  +  8  =  2a»-3. 
4  6 

6.   Solve   2x  =  4-f-. 


52. 

1.  Subtract  (a—b)x—(b  —  c)y  from  (a +5)  a? +(b-\-c)y. 

2.  Find  the  difference  in  value  between  the  arithmetical 

expression  57  and  the   algebraical   expression   ab, 
when  a  =  5  and  b  =  l. 

3.  Find  by  inspection  the  H.C.F.  of 

9(aV-4)  and  120V  +  4  a* +  4). 

4.  Simplify   (4  «-§)-*. 

5.  Solve  £±|-^lf=|. 

x-1        2x        3 

6.  There  is  a  rectangular  field  whose  length  exceeds  its 

breadth  by  16  yards.    The  field  contains  960  square 
yards.     Find  its  dimensions. 


ALGEBRA.  27 


53. 

1.   Find  the  L.C.M.  of 

4  (a3  -  ab2),    12  (ab2  +  b5),  and  8  (a"  -  a2  b). 

%   Simplify2^  +  2^  +  ^- 

3.  Simplify  (64<?16)-i 

4.  Divide    16  a;  —  y2  by  2#*  — yi 

5.  Solve  1--*±A  =  £zl6 

2z+l      a;  — 2 

6.  Find  two  numbers  in  the  ratio  of  2\  to  2,  such  that 

when  diminished  each  by  5  they  shall  be  in  the 
ratio  of  1 J  to  1. 


54. 

1.  Solve   x  +  2y=l 

y  +  2z=2 

3x  +  2y  =  z-l 

2.  The  sum  of  $330  is  laid  out  in  two  investments,  on  one 

of  which  15  per  cent  is  gained,  and  on  the  other  8 
per  cent  is  lost.  The  total  amount  returned  from 
the  investments  is  $345.     Find  each  investment. 

3.  Simplify    [(<*-»&*)*{-* 

4.  Divide   x~l  —  y~l  by  x~\  -  y~\. 

5.  Reduce  to  an  entire  surd  the  expression  ■£-(£)-*. 
2  1         3 


6.   Solve 


1      x-{-3     8" 


28  EXAMINATION    MANUAL. 


55. 

1.   Find  the  L.C.M.  of 

6(^y  +  a^),  9(V-:n/2),  and  4^  + ay*). 

x2        ,        x3 


2.  Simplify   -^ ^—  - 

1-s      (I-*)2  '  (l-*)3 

3.  Arrange  in  order  of  magnitude   2-^22,    3^7,    4V2. 

4.  Find  the  cube  root  of 

x«-  Sax"  +  5aV  -  3a5 a;  -  a6. 

5.  Simplify   3^432. 

6.  Solve  _^L_3*-2 

a;  +  4     2^-3 

56. 

1.  Show  that   V20,  V45,  VJ  are  similar  surds. 

2.  Simplify   2y/a~x  xVSaFb  X^JWx. 

3.  Simplify   -^128+^/686 +^16. 

4.  Simplify  ^2-8*  +  6  x  **-0*  +  20  x    **-7*. 

5.  Solve   i^2  +  H^  =  3i|. 

6.  If  each  side  of  a  square  plot  were  enlarged  by  10  yards, 

it  would  contain  1£  acres  more.     Find  the  dimen- 
sions of  the  plot.     (1  acre  =  4840  sq.  yds.) 


ALGEBKA.  29 


57. 

1.  Simplify  \ar%y{xff^(ar1y)'^\\ 

2.  Simplify    VT28  -  2 V50  +  V72  - Vl8. 

3.  By  selling  a  cow  for  $  24  I  lose  as  much  per  cent  as  the 

cow  cost  me.     What  was  the  cost  of  the  cow  ? 

1  Solve  m+it=-\ 

5.  Solve   x  +  y  +  z  =  5~\ 

x+y=z-7  >• 
x  —  3  =y-\-z) 

6.  A  person  has  travelled  3036  miles.     He  has  gone  7  miles 

by  water  to  4  on  foot,  and  5  by  water  to  2  on  horse- 
back.   How  many  miles  has  he  travelled  each  way  ? 


58. 

,    c  -,       27  —  x  ,  Sx  —  4     5x  —  2     0 

1.  Solve   -T-  +  —b —  =  2. 

2.  Find  by  inspection  the  L.C.M.  of 

a2-l,   a2  +  2a-3,   and   a?-7a2  +  6a. 

3.  A  steamer  takes  two  hours  and  forty  minutes  less  time 

to  travel  from  A  to  B  than  from  B  to  A.  The 
steamer  travels  at  the  rate  of  14  miles  per  hour  in 
still  water,  and  the  stream  flows  at  the  rate  of  If 
miles  per  hour.     Find  the  distance  from  A  to  B. 

4.  Which  is  the  greater  W%  or  £a/27  ? 

5.  Solve   !*_«-*«-- 


2x2 

6.  A  and  B  enter  into  partnership  with  a  joint  capital  of 
$  3400.  A  put  in  his  money  for  12  months  ;  B  put 
in  his  money  for  16  months.  In  closing  the  business, 
A's  share  was  $2070  including  his  profit,  and  B's 
share  $  1920.     Find  the  sum  put  in  by  each. 


30  EXAMINATION    MANUAL. 

59. 

1.  Solve   x  -f-  ay  =  b  ;     ax  —  by  =  c. 

2.  If  a  =  4,    &  =  3,    c=l,.  i=_7,   find  the  value  of 

a2  +  ac  +  52       V4a5  +  62  +  ^ c 

a?-ac  +  b2      -y/f^b^P^tid     a  +  b  +  c+d 

3.  Find  the  H.O.F.  of 

x'  +  Sx'-Qx-A   and   x* -Sx3  +  6x-4:. 

4.  Simplify 1 

x  —  y      x-\-y      x 

5.  Find  the  square  root  of  a2  —  4  ax  -f<6  a  +  4  #2  — 12 x  -f  9. 

6.  When  are  the  hour  and  minute  hands  of  a  clock  to- 

gether between  10  and  11  o'clock  ? 

7.  Solve  K*-l)  =  «*  +  l)--K*-l)2. 

60. 

1.   If  x  =  8,  y=  1,  find  the  value  of 


^{/+^S±Zj. 


j£ .  <y+y3 


2.  Simplify  -_^  +  2/2.    ^ 

3.  Find  the  H.O.F.  of 

xs-Sx2+lx-21  and  2#4+ 19  a;2 -f  35. 

4.  Find  the  square  root  of  {x2-\ 2a; ][!  +  -]  + 1. 

5.  Solve   ^x-i(8-x)-i(5  +  x)  +  ¥-  =  0. 

6.  Find  a  fraction  such  that  if  5  be  added  to  the  numer- 

ator, it  will  be  equal  to  1,  but  if  3  be  taken  from  the 
denominator,  it  will  be  equal  to  £. 

7.  A  boat's  crew  can  row  5  miles  an  hour  in  still  water. 

If  they  row  3£  miles  down  a  river  and  then  back, 
again  to  the  starting-point  in  1  hour  40  minutes, 
what  is  the  velocity  of  the  current?  . 


ALGEBRA.  31 


61. 

1.  Solve    7  a;2 -11a;  =  6. 

2.  Solve   _f^_^  =  -1A_.. 

a+b      a—b      a2—b2 

3.  Solve    -  +  ^=2;    bx-ay  =  0. 

a      b 

4.  Find  the  H.C.F.  of 

48a;2 -f  16a; -15  and  24a;3  -  22a;2  +  11  x-  5. 

5.  Divide    a?  +  b*  +  &  -Sabc  by  a  +  b  +  c. 

6.  Extract  the  square  root  of  x4,  —  2a;3  +  —  —  -  +  — 

H  2       2      16 

7.  A  vessel  containing  120  gallons  was  emptied  in  10  min- 

utes by  two  pipes  running  successively;  the. first 
pipe  discharged  14  gallons  in  a  minute,  and  the 
second  pipe  discharged  9  gallons  in  a  minute.  How 
many  minutes  did  each  pipe  run  ? 

62. 

1.  Subtract  a2 -13  ab +10  ac-c2  from  la2+12ab-3ac-b2, 

and  find  the  value  of  the  remainder,  when  a  =  1, 
5=2,  <?  =  -3. 

2.  Divide   x5 -.  by  x 

x5     J  x 

3.  Simplify  ^^^[;m2_j)(3^  +  J)] 

4.  Find  the  H.C.F.  of 

96a;4+8a;3-2a;  and  32a;3-  24a;2  -  8a;  +  3. 

5.  Extract  the  square  root  of 

a2-4a&2-6ac3  +  4£2(62  +  3c2)  +  9c6. 

6.  Solve    5a;  +  fy  =  52;     f  a;  +  4y  -  27. 

7.  Solve    5,  +  ^+g=10^-7  +  9. 

3a;-2       2s -1 

8.  When  are  the  hour  and  minute  hands  of  a  watch  at 

right  angles  between  3  and  4  o'clock  ? 


32  EXAMINATION   MANUAL. 

63. 

1.  Solve   ^Lii§+2^z3  =  9 

x  —  1       2a;  — 2 

2.  Solve   a;  +  2y  +  3z=101 

2a;  +  3y  +  4z  =  16  [• 
4a;  +  4y  +  5z  =  25  J 

3.  Solve    *f+l-Sx=^=±  +  3. 

3a;-2  4a;-3^ 

4.  Find  the  L.C.M.  of 

x3  -  4a;2  +  9a?  -  10  and  x3  +  2a;2  -  3a;  +  20. 
,    b  —  a 

5.  Simplify  y. -■ 

1      a(b  —  a) 

l  +  ba 

6.  A  and  B  have  the  same  income.     A  saves  -J-  of  his  in- 

come, and  B  spends  $50  a  year  more  than  A.  At 
end  of  the  4  years  B  finds  himself  $100  in  debt. 
Find  their  income. 

64. 

1.  Add  together 

1  x  -  (1  -  x)         and  10a;2  -x  -3 

1  ,       1         a;  +  3-(l-3a;y  4a;2+4a;  +  l' 

1  +  x 

2.  Divide  a;4  —  ±£x3  +  a;2  +  fa;  —  2  by  fa;  —  2. 

3.  Find  the  L.C.M.  of  a3  -  a2  b  -  ab  +  62  and  a4  -  62. 

4.  Extract  the  square  root  of  a2  -f-  b  to  three  terms. 

5.  Solve   aa;2  +  6a;-f  c  =  a(l  —  #)2  +  &(l  —  x)  —  c. 

6.  Solve   -JL-Jfi-i 

x-\-  i       a;       4 

7.  Two  numbers  consist  of  the  same  two  digits,  but  in  an 

inverse  order.  The  sum  of  the  two  numbers  is  55, 
and  their  difference  is  27.     Find  the  numbers. 


ALGEBRA.  33 


65. 


1.   Simplify - 

x-l  + - 


4  — a; 

2.  Find  the  L.C.M.  of  a;2- a;,  x2-l,  x*-2x2,  and  x*-4:X. 

o    a  -\    '    10a;  —  7      9a:  —  1      0 

3.  Solve -  =  2. 

2a;— 1       3  a; +  1 

4.  Solve   2a;  +  3y  +  4z  =  34;  3a;— 2y  +  2z=13;  y— 2=1. 

5.  Simplify   10a?-  {3a;- [- 4a;- (- 2a;  +  3a;)]J. 

6.  Solve   (a;  +  «)(>2-aa;  +  62)  =  a3+£a;2- 

66. 

1.    The  sum  of  two  numbers  is  1000,  and  their  difference  is 
■^  of  the  less  ;  find  them. 

„    a.      ,.,     4a;4—  1  K.Sx3  —  y 

3.  Find  the  H.C.F.  of  20a;*-a;2-l  and  25a;*-5a;3-a;-l. 

2       1 

4.  Extract  the  square  root  of  m2-f-2ra— 1  —  —  H =• 

5.  Solve    ***2LzJL=** 

6.  Solve  .  +  |„£. 

7.  When  a  term  is  transposed  from  one  side  of  an  equation 

to  the  other  side,  the  sign   of  the  term  must  be 
changed.     Why? 


34 


EXAMINATION    MANUAL. 


67. 


1.    Ify 


1 


and 


1 


1  +  z2  1  +  x 

of  x,  in  its  simplest  form. 

Add  together 

i-ft-(i-*)l,  ±-(*-5*), 

Simplify  {ap-q)p+q  X  (aq)q+r  -f-  (a?)p-q. 

x2  +  a2  +  b2  +  c2 


find  the  value  of  y  in  terms 


i-(i  +  4*)- 


Solve   x -\-  a -\- b -\- c 


a-\-  b  —  c-\-x 


Solve    *+2-2y 
3  J 

2X—  4:7/    , 


=  9. 


y 


23 
5   J 


6.  A  person  out  walking  has  18  miles  to  go,  and  finds  that 
at  the  rate  at  which  he  is  going  he  will  be  half  an 
hour  late,  but  if  he  quickens  his  pace  by  half  a  mile 
an  hour,  he  will  arrive  just  at  the  proper  time.  At 
what  rate  is  he  going  ? 


68. 

1.  Solve   ax2+b(2a  +  b)x-(ai-V)  =  0. 

2.  Multiply  x*  —  £as>f  f  by  $x  +  2. 

3.  Divide  x*  +  lOrr3  +  35tf2  +  24  +  50a:  by  (x  +!)(#  +  4). 


4,    Simplify 


x2  -f-  xy      x4,  —  y4 
x  —  y    '  (x  —  y)2 


5.    Solve    £+2-5=0 
3      5 

2*  +  |-17  =  0 
6.EXpand(|-MJ. 


ALGEBRA.  35 


69. 

1.  What  values  of  x  will  make  the  value  of  the  expression 

(x  —  a)  (x  —  b)  (x  —  c)(x  —  d)  equal  to  0  ? 

2.  Resolve  into  the  simplest  factors 

ra2  +  n2  -\-p2  +  q2  +  2(mq  —pn). 

ooi        o         #  +  3   ,   -,,-       12#  +  26 

3.  Solve   2x j- 1-  15  = -7 

3  5 

4.  Divide  $20,000  between  A,  B,  and  C  so  that  A's  share 

shall  be  i  of  C's,  and  A's  and  B's  together  shall  be 
equal  to  C's. 

5.  Find  the  H.C.F.  of 

6a3-  6a2y  +  2af-  2f  and  12a2-  15ay  +  Zy2. 

6.  Solve   ax2  —  a2(x-\-  b2)  =  ab(x  —  ab). 

70. 

1.  If  a  =3,  6  =  1,  c  =  -2,  d=0,  find  the  value  of 

a?-b3  .  g?+b2+c2+d2 

2ab+2b?+2ed+2ad  '  ^/(a-c}(b+c)+6(-c-bXa-2b') 

2.  Divide  (a2  +  2  be)3  -  4  be  (3  a'  +  4  b2  c2)  by  a2  -  2  be. 

3.  Find  the  H.C.F.  of 

6a3  — s* +  16  and  6x*-  3a;3  +  9x2  +  2>x+  12. 
1       1 

4.  Simplify  **  +  x  +}  x 


+  1 


x-      f-       ^  +  l  +  i 


5.    Simplify    


1      a?  +  l      x2  +  x+l 

6.  Solve   -  +  i=7;   -+i=6;    -+i=5. 

y^  a:       z  a;      y 

7.  Solve   J^  =  15_l 

07+  \      a;      4 


36  EXAMINATION   MANUAL. 


71. 

1.    Subtract  Bx-~-(2x  +  ^x2){rom^-^—(^-x\ 
8  2      6      \  4         / 

*   SimPlify4(IT^  +  4(I^)  +  2(IT^)' 

3.  Find  the  square  root  of 

(a  +  b)2-  c2  +  (a  +  c)2-  Z>2+  (6  +  c)2-  a2. 

4.  There  are  1200  books  in  my  library.     There  are  twice 

as  many  English  books  as  French  books,  and  five 
times  as  many  French  books  as  books  in  other  for- 
eign languages.  How  many  English  books  and 
how  many  French  books  do  I  have  ? 

5.  Solve   (4°'7rf£+1)-2«- 

4a2  +  b2 

6.  A  man  walking  -^  of  a  mile  above  his  ordinary  rate  gains 

f  of  an  hour  in  39  miles.  At  what  rate  does  he 
ordinarily  walk? 

72. 

1.  Divide  a"x  +  ax*  +  X*  by .-*-■ 

a3  —  x3  a  —  x 

2.  Extract  the  square  root  of  x2  +  £  to  three  terms. 

3.  Divide  ^— ^  by  — r 

Vc2  Vfl' 


4.    Solve  ^+y  +  2;  =  0;  a#  -f-  6?/  +  ez  =  0 ;  2#+3y  +  z  =  l. 

5i  Solve  ifo+ir-i- 

6.   Find  the  H.C.F.  of 

x"  +  5a:3—  7^2—  9x  —  10  and  2a;4  —  4a;3  +  8*  —  4. 


ALGEBRA.  37 


73. 


1.  Simplify  </a2bcX</ab^~a^b^cl 

2.  Find  the  L.C.M.  of  x2-  9x  +  20  and  x2  +  6  a?  —  55. 

3.  Solve     2^2  +  l  =  _g 1. 

9^2-16     4  +  3^     9 

4.  The  sum  of  two  numbers  exceeds  three  times  their  dif- 

ference by  8,  and  their  difference  is  equal  to  -J-  of  the 
smaller  number.     Find  the  numbers. 

5.  Solve  x  +  m~  2n^n  +  2m  —  2x 

x-\-m-\-2n     n  —  2m-{-2x 

6.  Extract  the  square  root  of 

8^2(2x2  -f  6ax  +  a)  +  c2(SQx2  +  12ar+  1). 

7.  Solve   (^+10J)2  =  4(^  +  9^). 


74. 

1.  Two  men  run  a  mile.     The  winner  finishes  in  4  minutes 

44  seconds,  and  wins  by  2  seconds.  How  many 
yards  start  must  the  loser  of  the  race  have  for  a  tie, 
if  each  runs  at  a  uniform  rate  ? 

2.  Find  two  consecutive  numbers,  the  difference  of  whose 

squares  is  51. 

3.  Solve  U12-x)  +  Sx-i  =  4x-i+H-x. 

5\  j  3  2  7 

4.  Find  the  H.C.F.  of  4^-f  4^2— %x—9  and  x3+2x2—x—2. 


Simplify  — —  +%' 

a  —  b      a-\-  b 
Solve    6O-l)O  +  i)  =  0. 


38  EXAMINATION   MANUAL. 


rye 

Find  the  value  of 

a  +VV  +  62       v  ,       -,  7 

when  a  =  —  4  and  o 


a3-2Z>(>2-62) 

2.  Bracket  together  the  equivalent  expressions  in  the  fol- 

lowing list: 

— ,  a2,  at,  a~z.   Va,  Va4,  a*,  vV,   — ,  a3  X  a-1. 
a3  a7 

3.  Divide  x* ;  bv  x 

#4     J  x  ■ 

4.  Solve   (s  +  5)(y  +  7)  =  (*+l)(y  +  9)  +  112) 

2tf+10  =  3y+l  j 

5.  Simplify  53A/'8^F  +  4a-v/^F2--^125a6612. 

6.  A  traveller  walks  a  certain  distance  at  a  certain  rate. 

If  he  had  gone  -J  a  mile  an  hour  faster,  he  would 
have  walked  the  distance  in  %  of  the  time.  If  he 
had  gone  ^  a  mile  slower,  he  would  have  been  2-J- 
hours  longer  on  the  road.  Find  the  distance,  and 
the  rate  at  which  he  travelled. 

7.  A  man  walking  £  of  a  mile  above  his  ordinary  rate  gains 

f  of  an  hour  in  39  miles.  Find  his  ordinary  rate  of 
walking. 

76. 

1.  Multiply  x%  —  4#*  +  2  by  x  —  xi. 

2.  Solve   _^i_=4+^^I 

■Vax  +  1  2 

3.  Show  that  if   -  =  -  =  -,  then  -  =  'y      — - 

J      d     /  o      xb  +  ya  +  zf 

4.  Divide   xiy-%  by  x~iyt. 

5.  Simplify  (3V8)(2V6). 

6.  Simplify  [\a\atff\*J. 

7.  Solve  a? —15f  +  — ^—  =  6. 

x  — 151 


ALGEBRA.  39 


77. 

1.  Two  ships  sail  at  the  same  time  from  the  same  port ;  one 

due  north  at  the  rate  of  9  miles  an  hour,  the  other 
due  east  at  the  rate  of  12  miles  an  hour.  After  how 
many  hours  will  they  be  60  miles  apart? 

0    a  ,         a(a—b).        .   ,  (b  —  a)b 

2.  Solve    — * 4- -\-a-\-b  —  x=  v  ,   ,    ' — 

x  —  a  —  o  a-f-  o  —  x 

3.  Solve    -=-^-. 

x     a —  x 


4.  Two  bodies  move  toward  each  other,  starting  from  points 

1800  feet  apart.  The  first  body  starts  5  seconds 
after  the  second  body,  and  the  two  bodies  meet  half- 
way between  the  points.  If  the  rate  of  the  first 
body  is  6  feet  a  second  more  than  that  of  the  second 
body,  find  the  rate  of  each  body. 

5.  Solve   9x2  —  9x  +  2  =  0. 

78. 

1.  Multiply  a2  —  a  +  1  —  a~x  by  a  +  1  +  a~l  +  a~\ 

2.  If  ?=  £  show  that  ^  =  £Z± 

o      a  c  +  a      c  —  d 

3.  What  is  the  value  of  [{^)t{-»? 

4.  Simplify  (a*  &"*)*+ (a~*  5s)*. 

5.  Divide  $  560  between  A  and  B,  so  that  for  every  dollar 

A  receives  B  shall  receive  $  2.50. 

6.  Solve   x2  —  ay  =  101 

xy  —  y2  =    6  J 

7.  For  what  positive  values  of  m  and  n  does  the  expression 

«» 
(—  a)  «  denote  an  imaginary  quantity  ? 


40  EXAMINATION    MANUAL. 

79. 

1.  Find  the  value  of  x*  ~  xy  +  ^  when 

x2  +  xy  +  y2 

V3  +  1        -,          V3  -  1 
x  — ■ —  and  ?/  =  — 

2  J  2 

2.  Find  the  L.C.M.  of  a-1,  1  -  a2,  a -2,  4 


3.  Solve    a  +  x+-V2ax  +  x2  =  b. 

4.  Find  two  numbers,  the  sum  of  which  is  15,  and  the  sum 

of  their  squares  113. 

5.  Two  freight  trains  pass  a  station  at  an  interval  of  4 

hours,  proceeding  at  the  rates  of  11  ?  and  17£  miles 
an  hour,  respectively.  At  what  distance  from  the 
station  will  the  faster  train  overtake  the  other  ? 

6.  Solve   ax2  -\-bx-\-c  =  0.     What  is  the  relation  between 

a  and  c  when  the  product  of  the  roots  of  this  equa- 
tion is  equal  to  unity? 

80. 

1.  Two  passengers  have  together  400  pounds  of  baggage. 

One  pays  $1.20,  the  other  $1.80,  for  excess  above  the 
weight  allowed.  If  all  the  baggage  had  belonged  to 
one  person  he  would  have  had  to  pay  $4.50.  How 
much  baggage  is  allowed  free  ? 

2.  Find  the  fourteenth  term  of  (2  a  -  b)l\ 

3.  Find  the  square  root  of 

16  m4  +  Jjf  m2n  +  8m2  +  |?z2  +  \n  +  1. 

4.  Find  the  H.C.F.  of  x"  -  ax3  +  a*x  -  a*  and  x3  -  a3. 


5.  Find  the  simplest  value  of  v64#3yV  +  -v#42/V2. 

6.  Solve  2bx  +  b_  1  1 


2bx-b      ax2 -a      2*2  +  *-l      2*2-3*+l 

7.    Find  two  numbers  whose  sum  is  7,  and  the  sum  of  their 
squares  exceeds  their  product  by  19. 


ALGEBRA. 


41 


81. 

1.    Find  the  value  of  ax  +  by  when 


91. 


br 


and  y 


ar  —  cp 


aq  —  bp~        "       aq  —  6^? 

2.  Add  *2-3^-fy2,  2^— §y3+z2,  37y-i2/2+2/3,  ^y-\if. 

0    0  ,        637+13       337  +  5       2^ 

3.  Solve    _^t —X^  =  _ 

4.  Solve    V4a;  +  9-2Va;  =  l. 

5.  Solve   ^+1_^±4£=7_ 9^+33 

7  14 

o      5a:  —  4?/             11  y—  19 
y  _3_. . — ^=37 z- 

u  2  4 

6.  Find  two  numbers  whose  difference  is  1,  and  the  sum  of 

their  squares  313. 

7.  Find  the  L.C.M.  of 

372  +  537  +  4,  ^  +  2*7-8,  and  372+737  +  12. 


82. 


37  (a  +  37) 


(f+12) 


1.  Solve   ^--337 

2.  The  difference  between  a  number  and  its  square  is  18 

more  than  the  same  difference  would  be  if  the  num- 
ber were  1  less.     Find  the  number. 

3.  Find  the  L.C.M.  of  a  -b,  a  +  b,  a2-b\  a?-b3,  az  +  b\ 

372  +  y2 

4.  Simplify 


~x        2        2 

v x       y  ■ 


l_l 

y      x 


373  +  y3 


5.  Solve   aa72  +  £37  +  c  =  0,  and  state  the  conditions  under 

which  the  values  of  37  will  be  (i.)  real,  (ii.)  equal, 
(iii.)  imaginary. 

6.  Find  the  product  of  2 2/3V23;3,   VQxy,  and  • 

7    ql  1  13  V3^ 

7.  bolve    37 =  y =  -• 

y     y     x     2 


42  EXAMINATION    MANUAL. 


83. 

1.  Find  the  H.C.F.  of 

x*-2a?  +  x2-8x  +  8  and  4:z3-  12^2  +  9^+  1. 

2.  Divide  -i^-  by   2Q^. 

25  ^"3        21^^a"2 

3.  A  can  do  a  piece  of  work  in  10  days  which  B  can  do  in 

8  days ;  after  A  has  been  at  work  upon  it  3  days,  B 
comes  to  help  him ;  in  what  time  will  they  finish  it? 

4.  Solve   ^±^-x-ZlA  =  a. 

x-2>     x  +  3 

5.  What  number  is  that  to  which  if  1,  5,  and  13,  severally, 

be  added,  the  first  sum  shall  be  to  the  second  as  the 
second  to  the  third  ? 

6.  Expand  (a  -3  x)\ 

84. 

1.  If   s  =  |  (a  -f-  b  -j-  c),  show  that 

s2  +  (s  -  a)2  +  0  -  b)2  +  (s  -  c)2  =  a2  +  b2  +  c2. 

2.  Subtract  L=L^  from 


l  +  3a;  1  — 3a; 

3.  A  man  has  6  hours  for  an  excursion.     How  far  can  he 

ride  out  in  a  carriage  which  travels  8  miles  an  hour 
so  as  to  return  home  in  time,  walking  back  at  the 
rate  of  four  miles  an  hour  ? 

4.  Kesolve  into  factors  x8  —  -^q. 

5.  Show  that,  if  \  =  %  then  7a  +  5ft  =  7g  +  5rf  ■ 

b      d  8a-Sb'     8c-Zd 


so- 


6.  Solve   x  —  y  =  \\     x2  —  y2  =  -£( 

7.  Solve    2*-«-2"     4a-7a 


bx  ax  —  bx      ab  —  b2 


ALGEBRA. 


85. 


43 


1.  Show  that  the  sum  of  the  squares  of  any  two  different 

numbers  is  always  greater  than  twice  their  product. 

«  '       ,  ,,    ,         ,  ,  4  +  12a?+93* 

2.  Reduce  to  its  lowest  terms        '      —  » 

2 -f- 13  a;  +  15  a;2 


3.    Find  the  value  of  a:  in 

96~3 

V5' 


\3**/         "'\4^tfP 


4.  Solve    V*(3  -  a?)  =  Va;  +  1  +  V2(ar-  1). 

5.  Extract  the  square  root  of  33  —  20  V2. 

6.  Expand  (l-^Y- 

86. 

,     ^,        a  —  2b  —  x        bb  —  x     .  2a— -a;— 195  _  q 
a2-4:b2       ax  +  2bx         2bx-ax 

2.   Simplify   1  + x 


I.  i       ZiX 

+  X  + 


3.    Solve 


1-x 
3a; -1      4a;-2      1 


2a;- 1      3.T-2      6 

4.  Expand  (2a;-32/)4. 

5.  A  vessel  which  has  two  pipes  can  be  filled  in  2  hours 

less  time  by  one  than  by  the  other,  and  by  both  to- 
gether in  2  hours  55  minutes.  How  long  will  it  take 
each  pipe  alone  to  fill  it  ? 


44  EXAMINATION    MANUAL. 


87. 

1.  If  a  =  2,  b  =  3,  x  =  6,  y  —  5,  what  is  the  value  of 

a-f  2a;-  j6  +  y-[a-a;-(6-2y)]|. 

2.  Simplify    |  V3  +  4  V=l  +^3  -  4  V^  i *• 

3.  Reduce  to  the  lowest  terms 

2a;2  -  8^  +  6       a:2-9a;  +  20    ,  2^2-10a: 
x2-bx  +  4:       x*-\0x  +  2l  '     x2-7x  ' 

4.  Solve    1 -f \ =1. 

x-{-  o  —  a      x  +  b  —  c 

5.  When  are  the  hands  of  a  clock  together  between  the 

hours  of  6  and  7  ? 

6.  The  plate  of  a  looking-glass  is  18  inches  by  12  inches, 

and  it  is  to  be  framed  with  a  frame  of  uniform  width, 
whose  area  is  to  be  equal  to  that  of  the  glass.  Find 
the  width  of  the  frame. 

7.  Show  that  a°=l. 

88. 

1.  Find  the  fourth  term  of  (2  a:  -  5y)12. 

2.  Divide  ? M-L-  by  ^M_^Zf. 

x     a-\-x     a  —  x         a  —  x      a-\-x 

3.  Solve   (a  —  b) x  =  (a -\- b)y;     x-\-y  =  c. 

4.  A  carriage,  horse,  and  harness  are  worth,  together,  $  720. 

The  carriage  is  worth  £  of  the  value  of  the  horse, 
and  the  harness  f  of  the    difference  between  the 
values  of  the  horse  and  carriage.    Find  the  value  of 
each. 
k    qi        x-\-13a-\-3b      -,  _a  —  2b 
5a  —  35  —  x  x-\-2b 


ALGEBRA.  45 


89. 

1.    Solve   xi-5xi  +  4:  =  0. 


0    o,  ,        ^2d1-xi  +  b^/2a-x      ^/a  +  b 

2.    Solve    —  J ==  —- 

V2a2-o;2-6V2a-a;      V a  -  b 

1 


3.   S^phfy   —X^x^ 
Mc-i 

4   Solve   2-1-1.    ?'+*-?• 

a      6  b      a      b 

5.  A  hare  is  50  of  her  leaps  before  a  greyhound,  and  takes 
four  leaps  to  his  three.  Two  of  the  greyhound's 
leaps  are  equal  to  three  of  the  hare's.  How  many 
leaps  must  the  greyhound  take  to  catch  the  hare  ? 

90. 

1.  Extract  the  square  root  of  81  a~%b$c. 

2.  Express  in  the  simplest  form  i  ( \/3  +  \  \/l92  +  \/81). 

3.  Simplify  {(a-^}~i. 

4.  If    a:b  =  c:d  =  e:f,  show  that  • 

a\b  =  aJrc-{-e:b-\-d  +/. 

5.  For  building  108  rods^of  stone-wall,  6  days  less  would 

have  been  required  if  3  rods  more  a  day  had  been 
built.     How  many  rods  a  day  were  built  ? 

6.  Solve  x-\-y  =  h\ 

x2-Sxy  +  4:y2  =  7)' 


46  EXAMINATION   MANUAL. 


91. 


1. 


*"*■*  cw 


2.  A  railway  train,  after  travelling  an  hour,  meets  with  an 

accident,  which  delays  it  1  hour ;  after  this  it  pro- 
ceeds at  f  of  its  former  rate,  and  arrives  5  hours 
late.  If  the  accident  had  occurred  50  miles  farther 
on,  the  train  would  have  been  3  'hours  20  'minutes 
late.     Find  the  length  of  the  line. 

3.  Prove  that  if  a  :  b  =  c  :  d,  then 

a-\-b  :  a  —  b  =  c-{-d:  c  —  d. 

4.  Solve  x~x  -f  ax~%  =  2  a2. 

5.  Write  the  5th  term  of  (3  a*  -  4y*)9. 

6.  What  is  a  surd  ?     Give  an  example.     What  are  similar 

surds?     Give  an  example. 


92. 

g&  qfl 

1.    Find  the  value  of  when  x  =  2? cfi  and  y=  2~*at- 


2,   Simplify 


x-y 

4a?-{-5y       4#  — 5y 


2x2-2xy     4(a;2-3/2)      ±{x-y)2 

a :**  (*)'*(^V(^. 

4.  Solve    VS  +  2a=iV£  +  4a 

Vaf+2  6      V*+3& 

5.  Find  a?  and  3/  from  the  proportions 

2a7  +  2/:y::  32/ :  2?/  —  a?j 

2x  +  l:y::2x  +  e>:y  +  2. 

6.  A  boat's  crew  row  3 \  miles  down  the  river  and  back 

again  in  1  hour  and  40  minutes.  If  the  current  of 
the  river  is  2  miles  per  hour,  determine  their  rate 
of  rowing  in  still  water. 


ALGEBRA.  47 


93. 

1.  Add ■_ 

V20a2m— 20a<?ra+5(?2ra  and  V20e2ra— Q0acm-\-4:ba2/m. 

2.  Find  the  H.O.F.  of 

4x2yXx2-\-xy+y2)\  §xy(x«-y«),  and  \&xz(x3~yi)'i. 

m    m  ■•     ,      _   _  ,   3a2 a?      6a2-a6-262      62a; 

3.  Solve    ate2  H — : 

1     $  c2  c 

4.  The  circumference  of  the  fore-wheel  of  a  carriage  is  9 

feet,  and  that  of  the  hind-wheel  12  feet.  What  is 
the  distance  gone  over  when  the  fore-wheel  has 
made  two  more  revolutions  than  the  hind- wheel  ? 

5.  Which  is  the  greater,  Vli  +  V7   or  Vl9  +  V2  ? 

Prove  your  answer. 

6.  Solve  x+y  =    6 1 

^  +  y3  =  72J  " 

94. 

1.  Solve  x2  +  xy-j-2y*  =  U) 

2:z2-:ry  +  3/2  =  16  J 

2.  Find  the  two  middle  terms  of  (  -  —  ^  ) . 

3.  Solve   2^2-21a;  +  55  =  0. 

4.  If  x'  and  #"  are  the  roots  of  the  equation  x2— px  +  q  =  0, 

show  that  #'  +  x"  =p,  and  #'#"  =  q. 

5.  Form  and  solve  the  quadratic  equation,  the  product  of 

whose  roots  is  42  and  their  sum  13. 

6.  A  number  having  two  digits  is  to  the  number  formed  by- 

interchanging  the  digits  as  7  is  to  4,  and  the  sum  of 
the  two  numbers  is  132.     Find  them. 


48  EXAMINATION   MANUAL. 


95. 

1.  For  what  value  of  m  will  the  equation  2#2  +  8;r  +  m  =  0 

have  equal  roots? 

2.  Find  the  value  of  a?+xy +y2  if  z  =  1+ V2,  3/  =  1- V2. 

3.  A  and  B  distribute  %  100  in  charity.     A  relieves  5  per- 

sons more  than  B,  and  B  gives  to  each  %  1  more  than 
A.     How  many  did  they  each  relieve  ? 

4.  Solve   x6-  20*3  =  189. 

5.  Multiply    (fly.    (^.    and  (£)* 

6.  Find  the  H.C.F.  of  . 

xi  +  xz-2x2+3x-S  and  at- ±x"-2x2  +  3x  +  2. 


96. 

t      w       vj     #3  —  8V3  —  z3  —  6#?/2: 
L   Simi>llfy      *3-^-2^     ' 

2.  Find  the  H.C.F.  and  the  L.C.M.  of 

34a2&ry,    27#y&3,    SlyW*. 

3.  Extract  the  square  root  of  a;4-) h  2aH — 

1  9T  s2        3        Sx 

4.  Solve   a:2  —  xy  =  f  1 

#y  +  y2  =  i  J 

5.  Solve   ax2-\-2bx-\-  c  =  0,  and  show  that  the  two  roots 

-  are  equal  if  b  is  a  mean  proportional  between  a  and  c. 


6.    Solve    V#  +  V4-f-#  =  ~ 


ALGEBRA.  49 


97. 


1.  Expand    (V^-^J- 

2.  Rationalize  the  denominator,  and  then  find  the  approxi- 

.        ,        ,  7  +  2  VlO 
mate  value  ot   — ! 

7-2VT0 

3.  Solve  15x-3x2+4:(x2-5x+&)$=-l. 

4.  A  man  starts  from  the  foot  of  a  mountain  to  walk  to  its 

summit.  His  rate  of  walking  during  the  second 
half  of  the  distance  is  £  mile  per  hour  less  than  his 
rate  during  the  first  half,  and  he  reaches  the  summit 
in  5 2  hours.  He  descends  in  3f  hours,  by  walking 
1  mile  more  per  hour  than  during  the  first  half  of 
the  ascent.  Find  the  distance  to  the  top,  and  the 
rates  of  walking. 

98. 

1.   Divide  2V3  +  3V2+ V30  by  3  V6. 

1.1,1 


2.   Simplify 


4(1  +  VV)      4(1  -Vx)      2(1  +  *) 

3.  Solve    V*  +  3  +  V*  +  8  =  5V*. 

4.  Find  the  mean  proportional  between  (a+5)2  and  (a— by 

5.  Solve       x  —  y  =  2        ) 

15(x2-y2)  =  lQxy)  ' 

6.  Expand  (2^?-^2)4. 


50  EXAMINATION   MANUAL. 

99. 

1.  Simplify  2 ^40  +  3^/108  +  ^5^-^320-2^1372. 

2.  Extract  the  square  root  of 

1  +  4y~*  -  2y-l  -  4y~x  +  25  y"*-  24y"*  +  16y~2. 

3.  Expand  [  — —  £yV# J  . 

4.  Solve   V2o;+l-V^=f4  =  iV^:r3. 

5.  Solve   ^±l_iA_2\  =  3^+lt 

6  x\b      a)  a 

6.  The  volume  of  a  sphere  varies  as  the  cube  of  its  diame- 

ter. Compare  the  volume  of  a  sphere  6  inches  in 
diameter  with  the  sum  of  the  volumes  of  three 
spheres  whose  diameters  are  3,  4,  5  inches,  respec- 
tively. 

100. 

i  i 

1.   Simplify 


•  Va2  —  x2      a  +  Va2  —  x2 

2.  Eationalize  the  denominator  of  -.  and  find  the 

3-2V2 
approximate  value  of  the  fraction. 

3.  Form  the  quadratic  equation  whose  roots  are  a  ±V—  56. 

4.  Solve   (or* +  2)(>-3 +  5)  =  or1 +  8. 

5.  A  vessel  which  has  two  pipes  can  be  filled  in  2  hours 

less  time  by  one  than  by  the  other,  and  by  both  to- 
gether in  1  hour  52  minutes  30  seconds.  How  long 
will  it  take  each  pipe  alone  to  fill  the  vessel  ? 


ALGEBRA.  51 


101. 

1.    Multiply  and  free  from  negative  exponents 
by  -  4  ar%  b~ V. 


2,  Solve  £±XEz£=| 

x  —  V9  —  x      3 

3.  Find  an  equivalent  expression  with  a  rational  denomi- 
V2l+  Vl2 


nator  for 


V7-V3 


4.  Solve   2x2  +  2>y2-xy  =  2>l')^ 

x~y  =    3J 

5.  Find  the  limit  of  the  series  1  +  $  +  £• 


6.  Reduce  to  partial  fractions  by  means  of  indeterminate 

rn    •        ,  6  a;2 4  #  —  6 

coefficients — —  • 

(x  -l)(x-  2)  (x  -  3) 

7.  Expand  (l-^Y. 

102. 

1.    Simplify  and  free  from  negative  exponents 
(-f  a-,58c-*rf-1)  X  (f  a2Z>-3<rW). 


2.  Multiply  -\/5-2^/6  by  3^4-^36. 

3.  Solve   x*-y3  =  e>3\ 

x  —  y  =    3  J 

4.  How  many  terms  of  the  series  54,  51,  48,  etc.,  amount  to 

513  ?     Explain  the  two  answers. 

5.  Solve    3a*-a;-i  +  2  =  0. 

6.  Find  the  first  five  terms,  and  the  (r  +  l)th  term,  in  the 

expansion  of  (1  —  x)~%. 


52  EXAMINATION    MANUAL. 

103. 


2.   Divide  x~6  +  y~'  by  x~*  +  y~3. 


3.   Solve   *  +  V2 


x  —  V2  —  a:2      o 

4.  A  debt  can  be  discharged  in  a  year  by  paying  $1  the 

first  week,  $3  the  second  week,  $5  the  third  week, 
etc.  Eequired  the  amount  of  the  debt,  and  the  last 
payment. 

5.  Find  the  coefficient  of  x12  in  the  expansion  of  (ab— &V)i 

6.  Solve  x2  +  xy  +  y2  =  52) 

xy  —  x2=    8 ) 

104. 

1.  Simplify  V5  X  ^2  X  </l. 

2.  Find  an  equivalent  expression  with  a  rational  denomi- 

4  +  V2 


nator  for 


4  +  V3 


3.  Solve   x2  —  2x+6^/x2  —  2a;  +  5  =  ll. 

4.  The  height  of  a  certain  triangle  is  4  inches  less  than  the 

base.  If  the  base  be  increased  by  6  inches,  and  the 
height  diminished  by  6  inches,  the  area  is  dimin- 
ished by  i.    Required  the  base  of  the  triangle. 

5.  Find  the  fifth  term  of  (3ar  -  2y)"10. 

x2 


6.    Resolve  into  partial  fractions 


(*»_!)(* -2) 


ALGEBRA.  53 

105. 

1.  Simplify  the  expression  "\-^_|_  £  x  \  Va  —  6. 

2.  Solve    — — ^ r  =  ^±^- 

rrvp  (x  +  «)       w  px 

3.  Solve    a;2  =  21  +  V(.*2  -  9). 

4.  Solve    bxy  =  84  —  a;2?/2 1 

x  —  3/=    6  J 

5.  The  sum  of  11  terms  of  an  arithmetical  progression  is 

22,  and  the  common  difference  is  -f.     Find  the  first 
term. 

6.  Find  the  coefficient  of  x9  in  (5  a3  —  4a;3)7. 

106. 

1.  Simplify  and  free  from  negative  exponents 

■(-  7  a-'b'c-5)  X  (Sa2b~sc). 

2.  Find  the  square  root  of  28  +  5  Vl2. 


3.  Solve    x2  —  X  +  5-V2X2  —  5x  +  6  =  J (3a; +  33). 

4.  Prove  the  formula  for  finding  the  sum  of  n  terms  of  an 

'  arithmetical  progression. 

5.  There  are  three  numbers  in  arithmetical  progression.   If 

1,  3,  9  are  added  to  them,  respectively,  they   are 
then  in  geometrical  progression.    Find  the  numbers. 

6.  Expand  to  five  terms  Vl  +  2x. 

7.  Give  an  example  of  a  series  of  terms  in  harmonical  pro- 

gression. 


54  EXAMINATION   MANUAL. 

107. 

1.  Multiply  5V14  +  3V5  by  7VI4-2V5. 

2.  Simplify  ^(a?b</tfxbxc)\ 

3.  Solve   ar\  +  2  =  x~\ +  8. 

ari  +  5 

4.  Solve   2^-3a:y  +  3/2  =  24\ 

3^-5^  +  2^  =  33]' 

5.  How  many  terms  of  the  series  19  +  17  +  15 amount 

to  91? 

6.  Prove  the  formula  for  finding  the  sum  of  n  terms  in 

geometrical  progression. 

7.  Find  the  fifth  term  in  the  expansion  of  Vl  +  x. 

108. 

1.  Divide  5-7V3  by  1+V3. 

2.  Find  an  arithmetical  progression  such  that  the  second, 

third,  fourth,  and  sixth  terms  may  form  a  proportion. 

3.  Solve   f-JU2* 

y      x       4 

3 

x  —  ?/  =  - 
y       2 

4.  A  person  bought  a  number  of  shares  in  a  railroad,  that 

cost  him  %  3000.  He  reserved  10  of  the  shares,  and 
sold  the  remainder  for  $2700,  gaining  $4  a  share. 
How  many  shares  did  he  buy  ? 

5.  Find  the  limit  of  the  sum  of  9  —  6  +  4  — 

1 


6.    Resolve  into  partial  fractions 


x*  —  cC 


7.    Find  the  fourth  term  in  the  expansion  of  [  5  —  -  j  • 


ALGEBRA.  55 


109. 


1.  Three  numbers  are  as  6  :  11  :  20.     If  1  is  added  to  each 

they    are   in    geometrical   progression.       Find   the 
numbers. 

2.  Solve  g  +  J  + «  =  **  +  <*&. 

a  -f-  b      b  bx 

3.  Divide  28  into  two  parts  such  that  their  squares  shall  be 

in  the  ratio  3  :  10. 

4.  The  first  term  of  a  series  in  arithmetical  progression  is 

17,  the  last  term  —  llf ,  and  the  sum  25^-.     Find 
the  common  difference. 

5.  Find  the  limit  of  the  sum  of  1  —  ■§■  -f-  YV  ~~ 

6.  Find  the  first  five  terms,  and  the  (r  -f  1)  th  term,  of  the 

expansion  of  (1  —  x2)~%. 


_         110. 

2.  Find  an  equivalent  expression  with  a  rational  denomi- 

,      ,     V20-V8 
nator  lor *-. 

V5+V2 

3.  Solve   ff-f-5  =  V#  +  5  +  6. 

4.  A  certain  capital  is  invested  at  4  per  cent.     If  the  num- 

ber of  dollars  in  the  capital  is  multiplied  by  the 
number  of  dollars  in  the  interest  for  5  months,  the 
result  is  117,0411.     Required  the  capital. 

5.  The  first  and  the  ninth  terms  of  a  series  in  arithmetical 

progression  are  5  and  22.  Find  the  sum  of  twenty- 
one  terms. 

6.  Find  the  middle  term  of  (1  +  x)2n. 

7.  Resolve  into  partial  fractions 

*2  —  9#-f-14 


56  EXAMINATION   MANUAL. 


111. 

2.  Find  the  H.O.F.  of 

^+7^+14^+5^-3  tmd2xi+9x*+8x2-x  +  e>. 

3.  A  merchant  buys  some  goods  for  $40,  and  sells  them  to 

another  merchant,  who,  in  his  turn,  sells  them  for 
$48£.  If  each  merchant  makes  a  profit  of  the  same 
rate  per  cent,  determine  what  that  rate  is. 

4.  If  a  :  b  —  c  :  d  =  e  :  /,  show  that 

a  +  2c  +  3e_2a  +  3c  +  4e 
b  +  2d+3f     26  +  3^+4/ 

5.  The  sum  of  five  terms  of  an  arithmetical  series  is  30,  and 

the  product  of  the  first  and  last  terms  is  20.  Form 
the  series. 

6.  If  a,  /S  are  the  roots  of  the  equation  x1  —  x  +  1  =  0,  show 

thata  +  £  =  l,  anda2  +  /?2  =  -l. 


-     112. 

1.  Solve   x*  +  3y2  =  28;    xy  +  y2  =  \2. 

2.  In  each  of  3  battles  36  officers  and  10  per  cent  of  the 

men  engaged  are  killed.  At  the  end  of  the  second 
battle  the  percentage  of  officers  to  men  is  two-thirds 
as  great  as  at  its  commencement.  The  number  of 
men  at  the  end  of  the  third  battle  is  equal  to  the 
square  of  the  number  of  officers  at  its  commence- 
ment. How  many  officers  and  men  were  engaged 
in  the  first  battle  ? 

3.  Find  an  arithmetical  progression  such  that  the  sum  of  n 

terms  shall  be  equal  to  n2. 

4.  Solve   z2  +  ^-2V2a;2  +  5*  +  3  =  ?. 

A  2 

5.  The  terms  of  a  ratio  are  7  and  3  ;  what  number  must  be 

added  to  each  in  order  that  the  ratio  may  be  halved? 


ALGEBRA.  57 


113. 

1.  Solve    ;r2  +  2y2  =  22;     2a;y-fy2:=21. 

2.  Simplify  2  V|  +  V60-Vl5 -Vf. 

3.  Solve    V9  x  +  40  +  2  Va;  +  7  =  Vx  +  44. 

4.  What  is  a  ratio  ?     Is  it  a  quantity  ? 

If  w  -f  w  :  m  —  n   =  x  -f-  y    :  a;  —  y,  show  that 
#2  -f-  ra2 :  a;2  —  m2  =  y2  +  n* :  y2  —  ?z2. 

5.  The  thickness  of  a  hollow  cylinder  varies  directly  as  the 

amount  of  material,  and  inversely  as  the  length  of 
the  cylinder  and  the  sum  of  the  radii  of  its  internal 
and  external  surfaces.  If  the  amount  of  material  in 
a  cylinder  50  feet  long,  whose  radii  of  external  and 
internal  surfaces  are  4  feet  and  3  feet,  respectively, 
he  1100  cubic  feet,  find  the  thickness  of  a  cylinder 
84  feet  long,  having  the  sum  of  its  radii  5  feet,  and 
containing  1650  cubic  feet  of  iron. 

114. 

1.  In  an  arithmetical  progression,  if  slf  s2,  s3  denote  the  sums 

to  the  nth,  2  nth,  3  nth  terms,  respectively,  prove  that 

53  =  3(52  —  8!). 

2.  In  a  geometrical  progression,  if  llf  l2,  l3,  denote  the  nth, 

2  nth,  3 nth  terms,  respectively,  prove  that  4s—  hb 

a r>  a        4.    -4.    i         4.4.  2x3—llx2y  +  l9xy2—10y3 

3.  Keduce  to  its  lowest  terms ^      '     * -f— 

x3  -7x2y  +13^y2  -6y* 

4.  Find  the  limit  of  1  +  4  +  i  + 

5.  A  committee  of  7  members  is  to  be  chosen  out  of  a  body 

of  20  protestants  and  15  catholics  in  such  a  way 
that  there  shall  be  3  protestants  and  4  catholics.  In 
how  many  different  ways  can  such  a  committee  be 
chosen. 


58  EXAMINATION    MANUAL. 


115. 

1.  If  A  varies  as-  B\  B*  as  C\  C5  as  Dr\  and  D7  as  E\ 

show  that  —  X  — -  X  —  X  — ,  does  not  vary  at  all. 
E     E     E     E  J 

2.  In  how  many  ways  can  2  white  balls  and  3  red  ones  be 

selected  out  of  an  urn  containing  7  white  balls  and 
10  red  balls? 

3.  Solve    — *■■ 1  =  1. 

y/x  —  V2  —  x      Va?  +  V2  —  x 

4.  Insert  10  arithmetical  means  between  6  and  61,  and  find 

the  sum  of  the  whole  series. 

5.  An  express-train,  travelling  at  uniform  speed,  after  be- 

ing an  hour  in  motion,  was  delayed  half  an  hour  by 
an  accident ;  after  which  it  proceeded  at  three-fourths 
of  its  original  rate  of  speed,  and  arrived  at  the  end 
of  its  journey  1  hour  and  50  minutes  late.  Had  the 
accident  occurred  after  the  train  had  travelled  a 
distance  of  60  miles,  it  would  have  been  1  hour  and 
40  minutes  late.     Find  the  length  of  the  line. 


116. 

1.  Find  an  arithmetical  progression  such  that  the  second, 

fourth,  and  eighth  terms  are  in  geometrical  progres- 
sion. 

2.  Solve   x2  —  ax-\-by\    if  —  ay  -\-  bx. 

3.  If  a  be  the  greatest  of  the  four  proportionals  a,  b,  c,  d, 

show  that  a—by-c  —  d. 

4.  If  four  numbers  be  in  proportion,  and  the  first  three  be 

in  arithmetical  progression,  show  that  the  reciprocals 
of  the  last  three  are  in  arithmetical  progression  also. 

5.  In  what  scale  of  notation  will  540  be  the  square  of  23  ? 


ALGEBRA.  59 


117. 

2.  Prove  that  a  ratio  of  greater  inequality  is  diminished  if 

the  same  quantity  be  added  to  both  its  terms. 

3.  Solve   4:X2  +  xy  =  Q;     Sxy  +  y2  =  10. 

4.  The  sum  of  10  terms  of  a  certain  geometrical  series  is  33 

times  the  sum  of  5  terms  of  the  same  series.  What 
is  the  common  ratio  ? 

5.  Sum  the  series  3,  2T77,  2T47,  ,  to  21  terms. 

6.  The  volume  of  a  sphere  varies  as  the  cube  of  its  radius, 

and  that  of  a  circular  plate  of  given  thickness  as  the 
square  of  its  radius.  If  the  volume  of  a  sphere  of 
radius  1  inch  be  equal  to  that  of  a  plate  of  radius  2 
inches,  find  the  radius  of  a  plate  which  is  equal  in 
volume  to  a  sphere  of  radius  4  inches. 

118. 

1.  In  an  arithmetical  series  the  common  difference  is  2,  and 

the  square  roots  of  the  first,  third,  and  sixth  terms 
also  form  an  arithmetical  series.     Find  the  series. 

2.  Find  a  geometrical   progression  such  that  the  sum  of 

an  infinite  number  of  terms  shall  be  4,  and  the 
second  term  shall  be  f . 

3.  What  is  the  equation  whose  roots  are  double  those  of 

the  equation  xA  +  x2  —  6  =  0  ? 

4.  A  cattle  dealer  buys  sheep,  and  sells  them  at  a  profit  of 

20  per  cent.  With  the  proceeds  he  again  buys  sheep, 
and  sells  them  so  as  to  gain  25  per  cent.  Once  more 
he  invests  the  proceeds  in  sheep,  and  this  time  he 
gains  16  per  cent.  If  his  last  profit  amounted  to 
$300,  how  much  money  did  he  invest  at  first  ? 

5.  Solve   x*  —  9f  =  as;     x  —  y  —  b. 


60  EXAMINATION   MANUAL. 

119. 

1.    Solve   0  +  a)5 -{x -of  =  242 a5. 


2.  Solve  x2 - x  +  3 VV -3#  +  36  =  2(>  +  26). 

3.  If  the  roots  of  tlie  equation  3  #2  —  8#  -f-  5  =  0  De  a  anc]  ^3 

show  that  the  equation  whose  roots  are  £  and  —  is 
15a;2  -Z4:x  + 15  =  0.  ^ 

4.  The  sum  of  three  quantities  is  y.     The  first  varies  in- 

versely as  x2,  the  second  inversely  as  x,  and  the  third 
is  constant.  When  #  =  4,  2,  1,  then  2/ =  3,  7,  21, 
respectively.     Find  the  equation  between  x  and  y. 

5.  Three  numbers  in  geometrical  progression,  if  multiplied 

by  3,  2,  and  1,  respectively,  are  in  arithmetical  pro- 
gression. If  the  middle  number  is  18,  what  are  the 
others  ? 

120. 

n     tc        -l      z.        -1/u       b  —  ca-\-b      a  —  bb-\-c 

1.  lia\o  =  o\c,  then  — = —  :  — ! —  = :  — ; — 

b  a  a  o 

2.  If  x  and  y  are  two  numbers,  A  their  arithmetical  mean, 

and  G  their  geometrical  mean,  then 
x2  +  y2  =  2(A2-G2). 

3.  If  z  vary  inversely  as  3x  -f-  £/,  and  y  vary  inversely  as  #, 

and  if,  when  x  =  1  and  y  =  2,  3  =  3,  find  the' value 
of  z  when  #  =  2. 

4.  Find  two  numbers  such  that  their  product,  their  sum, 

and  three  times  their  difference  are  in  the  projDortion 
5:2:4. 

5.  The  cost  of  an  entertainment  was  $120,  which  was  to 

have  been  divided  equally  among  the  party ;  but 
four  of  them  leave  without  paying,  and  the  rest  have 
each  to  pay  $2.50  extra,  in  consequence.  Of  how 
many  did  the  party  consist? 


ALGEBRA.  61 

121. 

1.  If  the  number  of  visitors  to  a  fair  varies  as  the  square  of 

the  number  of  degrees  above  42°  F.,  and  if  there  are 
1152  visitors  when  the  temperature  is  68°,  how 
many  visitors  will  there  be  if  the  temperature  is  55°  ? 

2.  Insert  between  6  and  16  two  numbers  such  that  the  first 

three  numbers  may  be  in  arithmetical  progression, 
and  the  last  three  in  geometrical  progression. 

3.  If  £  of  the  sum  of  the  squares  of  the  roots  of  the  equa- 

tion ax2  -f-  bx  +  c  =  0  is  equal  to  their  product,  find 
the  relation  between  a,  b,  and  c. 

4.  Solve   x-\-y=a\     x2-{-mxy-\-y2  —  b. 

5.  What  is  the  price  of  eggs  per  dozen  when  two  more  in  a 

dollar's  worth  lowers  the  price  one  cent  per  dozen  ? 

122. 

1.  Solve   x  —  y  =  a\     bx2  —  cx2  =  d. 

What  do  the  values  of  x  and  y  become  when  b  —  c? 

2.  The  side  of  a  square  is  a.     By  joining  the  middle  points 

of  its  sides  another  square  is  formed ;  by  joining  the 
middle  points  of  the  sides  of  this  square  a  third 
square  is  formed.  If  the  operation  is  continued  in- 
definitely, find  the  limit  of  the  sum  of  the  areas  of 
the  squares. 

3.  Separate  into  partial  fractions ^— t ^— 

r  ^  (x-l)(x-3)(x-5) 

4.  Find  the  fourth  term  of 


(d2-bx)$ 

5.  How  many  different  signals  can  be  made  with  ten  flags, 
of  which  three  are  white,  two  red,  and  the  rest  blue, 
if  all  are  hoisted  together,  one  above  another  ? 


62  EXAMINATION   MANUAL. 


123. 

1.  What  number  must  be  added  to  20,  50,  and  100,  respec- 

tively, that  the  results  may  be  in  geometrical  pro- 
gression ? 

2.  Find  an  arithmetical  progression  such  that  the  second, 

fourth,  and  eighth  terms  are  in  geometrical  progres- 
sion. 

3.  The  volume  of  a  sphere  varies  as  the  cube  of  its  radius. 

If  three  spheres,  whose  radii  are  9  inches,  12  inches, 
15  inches,  respectively,  are  melted  into  one,  what 
will  be  its  radius  ? 

4.  Solve   x2  —  y2~x2-}-yi  —  xy  =  3. 

5.  In  how  many  ways  can  a  base-ball  nine  be  arranged  if 

three  men  can  play  in  any  position,  and  the  others 
'  in  any  position  except  those  of  pitcher  and  catcher  ? 


124. 

1.  What  number  must  be  added  to  20,  50,  and  100,  respec- 

tively, that  the  results  may  be  in  harmonical  pro- 
gression ? 

2.  The  sum  of  four  numbers  in  geometrical  progression  is 

170,  and  the  third  exceeds  the  first  by  30.     Find 
them. 

3.  Solve   a?  — 11  ar8  +  37a-  —  35  =  0,  one  root  being  5. 


4.  Solve   4a?  +  4V33*-7ar+3  =  3ar(a?-l)  +  6. 

5.  Transform  3256  from  the  septenary  to  the  duodenary 

scale. 

6.  Two  steamers  ply  between  the  same  two  ports,  a  distance 

of  420  miles.  One  travels  half  a  mile  an  hour  faster 
than  the  other,  and  is  two  hours  less  on  the  journey. 
Find  their  rates. 


ALGEBRA.  63 


125. 

1.  In  an  arithmetical  series  the  second  term  is  21,  the  sev- 

enth term  41,  the  sum  1625.  Find  the  number  of 
terms. 

2.  The  first  and  the  seventh  terms  of  a  geometrical  series 

are  2  and  i.     Find  the  intermediate  terms. 

3.  Prove  that  the  difference  of  the  roots  of  the  equation 

x^—px  -f-  q  =  0  is  equal  to  the  difference  of  the  roots 
of  the  equation  x2  —  Spx  -f  2p2  -f-  q  —  0. 

4.  A  vessel  is  half  full  of  a  mixture  of  wine  and  water.     If 

filled  with  water,  the  ratio  of  the  quantity  of  water 
to  the  quantity  of  wine  will  be  9  times  as  great  as  if 
filled  with  wine.  Determine  the  original  quantities 
of  water  and  wine. 

1 


5.    Find  the  tenth  term  of 


126. 


1.  Jf  a :  b  =  c :  d  prove  that 

a  (a  +  b  +  c  +  d)  =  (a  +  b)  (a  +  c). 

2.  Solve    g  +  Va'-a^m 

x  —  Vcf-a?      n 

3.  Solve    rJ  +  y3  =  225yj 

3*-y»==    75    3 

4.  Insert  9  arithmetical  means  between  1  and  —  1. 

5.  There  are  four  numbers,  of  which  the  first  three  are  in 

geometrical  progression  and  the  last  three  in  arith- 
metical progression.  The  sum  of  the  first  and  last 
numbers  is  16,  and  the  sum  of  the  second  and  third 
numbers  is  12.     Find  the  numbers. 

6.  Find  the  fortieth  term  of  (1  —  x)~~°. 


64  EXAMINATION   MANUAL. 


127. 

1.  Find  an  equivalent  expression  with  a  rational  denomi- 

nator for t— £ 

3V2-2V3 

2.  Solve   f-i-tL^ia.       a?  +  y  =  12. 

3.  Two  vessels,  one  of  which  sails  faster  than  the  other  by 

2  miles  an  hour,  start  at  the  same  time  on  voyages 
of  1152  and  720  miles,  respectively.  The  slower 
vessel  reaches  its  destination  one  day  before  the 
other.    What  is  the  rate  per  hour  of  the  faster  vessel  ? 

4.  There   are   five   numbers   in    arithmetical    progression. 

Their  sum  is  to  the  sum  of  their  squares  as  9  :  89. 
The  sum  of  the  first  four  numbers  is  32.  Find  the 
numbers. 

5.  Find  the  limit  of  the  sum  of  4  +  3  +  f  + 

6.  Find  the  value  of  x  in  an  infinite  series  in  terms  of  y,  if 

y=l  —  2x  +  2>x\ 

128. 

1.  Eeduce  to  the  simplest  form 

2  \/3  (v^9  -  2  </M  +  4  -\/J  -  3  </2). 

2.  Solve   3xt-x~i+2  =  0. 

3.  In  t  seconds  a  body  falling  freely  describes  16  f  feet. 

The  velocity  of  sound  is  1100  feet  a  second,  very 
nearly.  If  a  stone  dropped  from  the  top  of  a  tower 
is  heard  to  strike  the  ground  after  4  seconds,  find 
the  height  of  the  tower. 

4.  The  sum  of  three  numbers  in  arithmetical  progression  is 

24,  and  their  product  is  480.     Find  them. 

5.  The  third  and  the  seventh  terms  of  a  geometrical  series 

are  12  and  192.     Find  the  tenth  term. 

6.  Find  the  tenth  term  of  — — —  • 

(1  —  bxy 

7.  In  what  scale  is  4954  expressed  by  20305  ? 


ALGEBRA.  65 


129. 

1.  Find  the  arithmetical  mean,  the  geometric  mean,  and 

the  harmonic  mean  between  a  +  b  and  a  —  b. 

2.  A  man  divided  $216  equally  among  a  certain  number 

of  persons.  If  there  had  been  three  more  persons, 
each  would  have  received  $1  less.  Required  the 
number  of  persons. 

3.  Solve   #*  +  #*  =  706;       x  —  y=2. 

4.  Find   the   value  of  the   last   term    of  an  arithmetical 

series,  having  given  the  first  term,  the  common  dif- 
ference, and  the  sum  of  the  series. 

5.  Develop  (3  or1  —  2  a:)-4  to  five  terms. 

6.  Separate  into  partial  fractions  — — ■ ■ -^— 

a?  —  bxl  —  x-\-  30 

7.  How  many  different  arrangements   can  be  made  with 

the  letters  of  the  word  freshman,  if  the  letters  mna 
are  never  separated  ? 

130. 

1.  Divide   iVF.by  V2  +  3vT 

2.  Solve   *  +  4+^+iY=-12r 

\x—y       x—A 

3.  A  sculptor  purchased  two  cubical  blocks  of  marble  for 

$2960,  at  $5  per  cubic  foot.  The  length  of  the  two 
together  was  12  feet.     Find  the  length  of  each. 

4.  An  army  on  'the  march  is  advancing  at  the  rate  of  12 

miles  a  day,  when  a  detachment  55  miles  in  the 
rear  is  ordered  to  join  it.  How  long  will  it  take  to 
do  so,  if  it  can  advance  25  miles  the  first  day,  24 
the  next,  23  the  next,  and  so  on? 

5.  The  third  term  of  a  geometric  series  is  20,  the  eighth 

term  is  640,  and  the  sum  of  the  series  20475.  How 
many  terms  are  there  ? 

6.  Insert  five  harmonic  means  between  —  1  and  h 

7.  What  is  the  sixth  term  of  (a2-  x2)~2? 


EXAMINATION    MANUAL. 


131. 

1.  Solve    9rr-3.?2  -f  4(s»  — 3s  +  5)*  =  11. 

2.  Solve   x2  +  a?y  +  y2  =  52 ;       ^y  —  s8  =  8. 

3.  From  two  towns,  168  miles  apart,  A  and  B  set  out  to 

meet  each  other.  A  went  3  miles  the  first  day,  5 
the  second,  7  the  third,  and  so  on.  B  went  4  miles 
the  first  day,  6  the  second,  8  the  third,  and  so  on. 
In  how  many  days  did  they  meet? 

4.  The  sum  of  three  numbers  in  harmonical  progression  is 

37,  and  the  sum  of  their  squares  is  469.  Find  the 
numbers. 

5.  Expand  to  six  terms,  in  a  series  of  ascending  powers  of  x, 

the  fraction  - — ^— 

1  —  x  —  x2 

6.  How  many  numbers  between  10,000  and  15,000  can  be 

formed  with  the  digits  0,  1,  2,  3,  4,  5,  6,  no  repeti- 
tions being  allowed,  and  each  number  to  be  divisi- 
ble by  5? 

132. 

1.  Bationalize  the  denominator  of  the  fraction  -^— — 

2.  Solve   a*  +  5a* -22  =  0.  V5+V2 

3.  Solve    (^  +  .?/)2  +  (^-:?/)2  =  8;     a;2  +  ?/2  =  2(a2  +  &2). 

4.  There  are  two  arithmetical  series  which  have  the  same 

■#  common  difference.  The  first  terms  are  3  and  5, 
respectively,  and  the  sum  of  seven  terms  of  one  is 
to  the  sum  of  seven  terms  of  the  other  as  2:3. 
Determine  the  series. 

5.  What  is  the  least  integer  which  added  to  the  ratio  9  :  23 

will  make  it  greater  than  the  ratio  7  :  11  ? 

6.  Ten  papers  are  to  be  set  at  an   examination,  four  of 

them  being  in  mathematics.  In  how  many  differ- 
ent orders  can  they  be  set  so  that  the  mathematical 
papers  shall  always  come  together  ? 


ALGEBRA.  67 


133. 

1.  Solve   2x-y  =  2;     8x3-y3  =  98. 

2.  A  and  B  start  at  the  same  time  from  the  places  C  and 

D,  A  to  travel  from  C  to  D,  B  to  travel  from  D  to 
C.  When  they  meet,  A  had  travelled  thirty  miles 
more  than  B.  If  they  should  continue  at  the  same 
rates,  A  would  finish  his  journey  in  4  days  and  B 
in  9  days.     Find  the  distance  from  Cto  D. 

3.  The  first  term  of  an  arithmetical  progression  is  f ,  and 

the  difference  between  the  third  and  the  seventh 
terms  is  3.     Find  the  sum  of  n  terms. 

4.  Find  the  limit  of  the  sum  of  f  —  \  -f  f  — 

5*  Resolve  into  partial  fractions 


^-7^+36 

6.  From  a  society  consisting  of  12  men  and  8  women,  in 
how  many  ways  can  a  committee  of  4  men  and  3 
women  be  selected?  In  how  many  of  these  ways 
will  a  particular  woman  always  be  included  ? 


134. 

1.  If  a  varies  directly  as  V&,  and  inversely  as  c3,  and  if 

a  =  3  when  b  =  256  and  e  ==  2,  find  b  when  a  =  24 
and  c  =  J. 

2.  Solve    ■v'a?  +  22 -:-&e+1j  =  1. 

3.  Solve    o;2  +  ?/2  +  ^  +  y  =  78;     a;  +  y  +  ay  =  39. 

4.  Find    four   numbers   in    arithmetical  progression,  such 

that  the  sum  of  the  squares  of  the  first  and  the 
second  shall  be  29,  and  the  sum  of  the  squares  of 
the  third  and  the  fourth  shall  be  185. 

5.  Insert  three  geometrical  means  between  \  and  128. 

6.  Fin  d  the  sixth  term  of-  (4  a2  ex  —  3  0$yi)i 


68  EXAMINATION    MANUAL. 

135. 

1.  Simplify  V2a/2V1-a/  V^l. 

2.  Solve  xn  —  axln  =  b. 

3.  A  stone  is  dropped  from  the  top  of  a  tower,  and  when  it 

has  fallen  exactly  half-way  to  a  window  a  feet  below 
the  top,  another  stone  is  dropped  from  the  window. 
After  how  many  seconds,  reckoned  from  the  time 
when  the  first  stone  is  dropped,  will  the  first  stone 
overtake  the  second?  The  space  described  by  a 
falling  body  in  t  seconds  is  hgi1. 

4.  The  difference  between   two  numbers  is   48,  and  the 

arithmetical  mean  exceeds  the  geometrical  mean  by 
18.     What  are  the  numbers  ? 

5.  Find  the  value  of  x  in  an  infinite  series  in  terms  of  y,  if 

y  =  1  -f  x  —  2x2  -f  x\ 

6.  A  boat's  crew  consists  of  8  men ;  2  men  can  row  only 

on  the  port  side,  and  2  others  only  on  the  starboard 
side.     In  how  many  ways  can  the  crew  be  arranged? 


136. 

1.  Solve   ay2  -f  hxy  —  5  =  0;     bx2  -f-  axy  —  a  =  0. 

2.  Find  in  the  simplest  form  the  value  of  (1  +  V —  3)6. 

3.  If  the  second  term  of  a  geometric  series  is  24,  and  the 

fifth  term  81,  find  the  series. 

4.  If  a  :  b  =  c  :  d,  prove  that  Va  +  b  :  V3  =  Vc  +  d :  V5. 

.5.  On  how  many  nights  may  a  different  guard  of  5  men 
be  selected  from  a  company  of  36  men,  and  on  how 
many  of  these  would  the  oldest  man  have  to  serve  ? 

6.   Ify  =  #  —  —  -f— ,  find  the  value  of  a;,  in  terms  of  y, 


ALGEBRA.  69 


137. 

1.    Find  the  three  cube  roots  of  1. 


,    Solve  ^+^  =  2;     ^_y  =  75. 

3.  How   many  signals   can   be  made  with  3  blue  and  2 

white  flags  which  can  be  displayed  singly,  or  any 
number  at  a  time,  one  above  another  ? 

4.  Find  the  least  number  which,  if  divided  by  15,  leaves 

a  remainder  of  14,  and  if  divided  by  13,  leaves  a 
remainder  of  12. 

5.  Find  by  the  method  of  finite  differences  the  sum  of 

3  +  11  +  31  +  69  +  131  + ,  to  20  terms. 

6.  How  many  balls  are  there  in  an  incomplete  square  pile 

if  the  upper  course  consists  of  529  balls  and  the 
base  of  5184? 

138. 

1.  Solve   x2  -f  y2  =  axy ;     x  -f  y  =  bxy. 

2.  From  a  horse-car  station,  a  closed  car  leaves  every  9 

minutes,  beginning  at  7  o'clock  a.m.,  and  an  open 
car  every  16  minutes.  At  what  times  will  an  open 
car  leave  exactly  3  minutes  after  a  closed  car  ? 

3.  If  on  an  average  A  solves  3  problems  out  of  5  which 

he  tries,  and  B  solves  2  out  of  5,  what  is  the  proba- 
bility that  a  problem  which  both  try  will  be  solved  ? 

4.  Find   by  the  method  of  indeterminate  coefficients  the 

sum  of  1  +  23  -f  33  -f  43  + ,  to  twenty  terms. 

5.  If  y  =  x  -f  x2  -f  or3,  find  the  value  of  x  in  terms  of  y,  de- 

veloping the  series  to  six  terms. 


70  EXAMINATION   MANUAL. 


139. 

1.  Solve    _  +  _=— ^;       _+  _=  _. 

x     y     x  +  y        x2     f     a? 

2.  A  man  wishes  to  make  up  as  many  different  parties  as 

he  can  out  of  20  friends,  each  party  consisting  of 
the  same  number.  How  many  should  he  invite  at 
a  time,  and  how  many  parties  will  there  be  ? 

3.  If  on  an  average  2  ships  out  of  5  return  safe  to  port, 

what  is  the  chance  that  out  of  5  ships  expected  at 
least  two  will  return  ? 

4.  A  man  wishes  to   build  a  wall  31  feet  long,  and  has 

stones  of  4  different  lengths :  viz.,  2.2  feet,  2.5  feet, 
3  feet,  4.1  feet.  How  many  of  each  kind  may  he 
use  in  laying  a  course  ? 

5.  Solve  by  resolution  into  factors  x*  —  3x2-\-  Sx  —  1  =  0. 

6.  Find    by   continued    fractions    the   fourth   convergent 

value  for  VTO.  . 


140. 

1.  What  relation  exists  between  the  two  roots  of  the  equa- 

tion ax2  -f-  bx  -f-  a  =  0? 

2.  Solve  a*— f=S\     O  +  y) (**  +  f)  =  7. 

3.  Find  by  means  of  the  binomial  theorem  the  value  of 

V16.16  correct  to  five  decimal  places. 

4.  Find  by  the  method  of  finite  differences  the  sum  of  the 

infinite  series  1  +  bx  +  9x2  +  13*3  +  llx4  + 

5.  Expand  to  four  terms  by  the  method  of  indeterminate 

coefficients ■ — -• 

1  +  x  +  x2 

6.  Find  the  least  integral  values  of  x  and  y  which  will 

satisfy  the  equation  x2  —  52?/  =  1. 


ALGEBRA.  71 


141. 


1.  Define  a  logarithm,  its  characteristic,  its  mantissa.    Prove 

that  logmn  =  logm  +  logn. 

2.  If  2  is  the  base  of  a  system  of  logarithms,  find  the  loga- 

rithms of  8,  -^j,  and  Vl6. 

3.  Given  loga,  log b,  logo,  and  logo?;  how  is  the  value  of 

log —  found  ? 

4.  Find  by  logarithms  the  value  of  24.13  X  6.052. 

5.  Find  by  logarithms  the  cube  root  of  3852. 

6.  Find  the  value  of  8352x3.69 

(30.57)3 


142. 

1.  State  and  prove  the  rule  for  finding  the  logarithm  of  a 

quotient. 

2.  If  the  base  of  a  system  of  logarithms  is  3,  find  log  9, 

logA.,log27Mog^S. 

3.  What  is  colog  10  in  the  common  system  ? 

4.  If  log  6492  =  3.81238,  find  log  0.00006492. 

5.  Find  the  mean  proportional  between  (0.01)*  and  (0.2)4. 

6.  Find  the  value  of  (0-005234)*  x  (0.017)*. 

24* 


72  EXAMINATION   MANUAL. 


143. 

1.  "What  is  the  logarithm  of  1  in  any  system  ?  of  the  base 

of  the  system  ?  of  the  reciprocal  of  the  base  ?  If  the 
base  is  greater  than  1,  for  what  numbers  are  the  loga- 
rithms positive,  and  for  what  negative?  Give  rea- 
sons for  your  answers. 

2.  If  the  base  of  a  system  of  logarithms  is  J- ,  find  log  1^-}. 

logW^ndlog-^S- 


3.  If  log  a  =  0.78241,  and  log  b  =  0.63575,  find  logVa2+Z>2. 

4.  Find  the  value  of   , 

-\/0472 

5.  Solve  20*  =100. 

144. 

1.  If  the  base  of  a  system  of  logarithms  is  64,  find  log  4, 

logl6,  log32,  log^. 

2.  Of  what  number  is  —  h  the  logarithm  in  a  system  whose 

base  is  16  ? 

35*  X  12" 


3.    Find  the  value  of 


13 


a     xv   A  .,  .    f  VO0125  X  V31.15 

4.  Imd  the  square  root  ol ttt^t^j 

0.00081 

5.  Solve  10*  =  2.45. 

6.  A  man  owes  $14,720.20.     At  the  end  of  each  year  he 

pays  §2000.      In  how  many  years  will  the  debt  be 
paid,  the  rate  of  interest  being  6  per  cent  ? 


ALGEBRA.  73 


145 


1.  If  the  base  of  a  system  of  logarithms  is  —  f,  find  log— ^-, 

log  |,  log  ft. 

2.  Of  what  number  is  —  5  the  logarithm  in  the  common 

system  ?  in  the  system  whose  base  is  3  ?  in  the  sys- 
tem whose  base  is  $  ? 

3.  For  what  base  is  log  1  ^  0  equal  to  4  ? 

4.  Find  the  sixth  root  of  0.000000004096. 

5.  Find  the  value  of  *2  X  (O0016)*. 

a/108 

6.  Solve  (!•)*=  17.4. 


146. 

1.  If  the  base  of  a  system  of  logarithms  is  —  6,  find  log  36, 

log  1296,  log  ~*fr. 

2.  Find  the  value  of  (8.31)-°-27. 

3.  If  </M7  :  (0.84)2  =  -^0.054321  :  x,  find  the  value  of  x. 

4.  Find  the  value  of  (15-6)»  X  (0.0015)*. 

(0.00065)* 

5.  Solve  (i)»  =  i, 

6.  What  sum  will  amount  to  $1000  in  10  years  at  5  per 

cent  per  annum  compound  interest  ? 


74  EXAMINATION    MANUAL. 

147. 

1.  What  is  the  logarithm  of  512  to  the  base  2  V2? 

2.  What  is  the  base  of  the  system  in  which  log  81  =  —  4  ? 

.    „.    ,  .,         ,        ,  1.265  X  0.01628 

3.  Fmd  the  value  of  —^-^j— 

4.  Find  the  mean  proportional  between 

V4?756and  ^(O0078)"2. 


5.  Find  the  value  of  \381  +  V58. 

6.  In  how  many  years  will  a  given  sum  of  money  treble 

itself,  at  3  per  cent  per  annum  compound  interest, 
payable  semi-annually? 


148. 

1.   What  is  the  logarithm  of  T^  to  the  base  2  V3? 


2.  Find  the  value  of  V(24.4)"*. 

3.  Given  the  Napierian  base  2.71828;  find  the  Napierian 

logarithms    of  19,  23,  29,  and  31,  to  five  decimal 
places. 

4.  Find  the  value  of  (   .  )  • 

W0.01/ 

5.  A  man  places  $25,000  at  5  per  cent  compound  interest, 

and  draws  out  §  1000  at  the  end  of  each  year.    What 
will  be  due  at  the  end  of  12  years? 

6.  Solve   2*  =  769. 


ALGEBRA.  75 


149. 

1.  What  is  log  1  in  any  system  ?     Why  ? 

2.  If  the  base  of  a  system  of  logarithms  is  a,  what  is  log  a, 

log-,  log  a3,  log^a7,  log—,  logf- 

3.  Find  the  value  of  A/^4  +  \/5. 


9347x-n/0.0073 


4.  Find  the  value  of 

5.  Given  the  Napierian  base  e  =  2.71828,  loge5  =  1.60943, 

loge  11  =  2.39789;    find  to  five  decimal  places  the 
common  logarithms  of  5  and  11. 

150. 

1.  If  the  base  of  the  system  is  am,  what  is  log  am,  log  a5™, 

1  Cbm 

log-,  log  1,  log  a™,  log—? 

2.  Find  the  value  of  4^5. 

^0.01 

A^69  +  3  -yTl9 


3.  Find  the  value  of 

18V0.95 

4.  Find  the  modulus  for  changing  common  logarithms  to 

those  whose  base  is  7. 

5.  Find  to  five  decimal  places  log  54  in  a  system  whose 

base  is  12. 

6.  A  spendthrift,  at  the  age  of  30,  comes  into  possession  of 

a  fortune  of  $200,000,  which  pays  5  per  cent.  He 
spends  each  year  all  his  income,  and  also  borrows 
$2000  at  5  per  cent  compound  interest.  How  old 
will  he  be  when  his  fortune  is  all  gone  ? 


EXAMINATION  PAPEKS  IN  AIGEBEA. 


I. 

Bowdoin  College,  Brunswick,  Me. 

Examination  for  Admission,  June,  1883. 

1.  Divide  a^ar5 +  5^-4^ -3  by  x2-2x-2>. 
Divide  a6  -  b«  by  a3  +  2a2b  +  2ab2  +  b\ 

2.  Resolve  x16  —  3/16  into  five  factors. 

3.  Find  the  L.C.M.  of  (x  +  2a)3,  (x-2af,  and (x2 -  4a2). 

4.  From subtract 


a  —  x  a?  —  x2 

5.  Multiply  together  i^^2,   — ~-£,    andl+— £-• 

1  -j~  y     x  -j-  x  J.  —  x 

6.  (*  +  f)(*-f)  +  t  =  (*7  +  5)(*-3).     Finda;. 

7     437  +  81       R      12a? +  97      i'    if.  %  -, 

'•   vk TTr  ~  «•     tt — ^t;  —  4.     Find  a;  and  y. 

lOy  — 17  15y-17  y 

8.  Find  the  square  root  of  x*  —  x3  +  —  +  4  ar  -  2  +  —^ 

9,  —^ Reduce  this  to  a  fraction  with  ra- 

1  +  a-Vl-a2 

tional  denominator. 

10.    Ja?-ia?  +  7i=8.         Find  a:. 
.r6  -  8ar»  =  513.     Find  ar. 


78 


EXAMINATION    PAPERS. 


II. 

Dartmouth  College,  Hanover,  KH. 

Entrance  Examination,  June,  1883. 

1.  Divide  a2  +  — 2  —  2  by a. 

a2  J  a 

2.  Resolve  a12  —  x12  into  six  factors. 

3.  Find  the  L.C.M.  of  a?  -  x,  x2  -  x  -  2,  and  r>  +  1. 

4.  Reduce  f    „       -2  —     4_    4  )  (y2  +  ar2)  to  its  sinplest  form. 

* q — )  =  ~4T' 

3 

(iii.)    • 


(ii.) 


x     y 

2  +  4 


1      1_1 

x^~  y~~  4 


2*    y_9 


a?     y~12 


6.    Write  the  values  of  8"!,  8°,  16*,  and  (8a*£2)i 
It  Multiply  together  Vabc2,  aib^cb,  and  a'SbSc1. 
8.    Divide  a3  -  b2  by  a*  -  V&. 


1.    Multiply  together 


III. 

Boston  University,  Boston,  Mass. 

Examination  for  Admission,  June,  1882. 
I-*2     l-v2         *       1 


-£-,  and 


id  sim- 


a; 


1+2/     a;  +  x2  1 

plify  the  result  as  much  as  possible. 

0    n.  y  —  'z      x  +  z      1       x  —  y      x  — 

2.    Given   ^— -+-  —  -»      — r-^ r 

2  4         2  5  6 

2HL?  =  a?  +  .V  _  4.        Find  the  values  of  #,  y,  and  z. 


=  0,      and 


ALGEBRA.  79 


3.  Simplify   _JL. -_£-  +  ;£_. 

4.  Extract  the  square  root  of  40  x  +  25  — 14  ar*  +  9  x*  -r-  24  s*. 

5.  Given  3 a?2  —  2sy  =15;  2#  +  3y  =  12.     Find  values  of 

a;  and  y. 

v^X  a/3 

6.  Express = with  a  single  radical  sign. 

V2 

7.  Expand  (1  —  2x2f  by  the  binomial  theorem. 


IV. 

Brown  University,  Providence,  R.I. 

Examination  for  Admission,  June,  1883. 

1.  Factor  x*  —  y* ;    also,  factor  4  a4  —  8  a3  #  +  4  a2  a8. 

0    6#+7      2#-2     2#  +  l      «  ■,      ,        £ 

2.  — — ; -r  =  — -i —      Find  value  of  x. 

15  7x  —  6  5 

3.  A  sum  of  money  is  divided  equally  among  a  certain 

number  of  persons;  if  there  had  been  four  more, 
each  would  have  received  a  dollar  less  than  he  did ; 
if  there  had  been  five  fewer,  each  would  have  re- 
ceived two  dollars  more  than  he  did ;  find  the  num- 
ber of  persons  and  what  each  received. 

4.  Multiply  a§  -  c&  +  1  -  or  I  +  cr\  by  oh  +  1  -  or*. 

5.  Va-frz  +  Va  —  x  =  V&-     Find  value  of  x. 

6.  ;r  +  y  =  4;     --f.-s-l.     Find  values  of  x  and  y. 

x    y  .  . 

7.  A  boat's  crew  row  3 }  miles  down  a  river  and  back  again 

in  1  hour  and  40  minutes ;  supposing  the  river  to 
have  a  current  of  2  miles  per  hour,  find  the  rate  at 
which  the  crew  would  row  in  still  water. 

8.  Find  sum  of  six  terms  of  the  geometrical  progression  of 

which  -§  is  the  first  term  and  •§  the  second  term. 


80  EXAMINATION    PAPERS. 

V. 

Mass.  Institute  of  Technology,  Boston,  Mass. 

Entrance  Examination,  June,  1882. 

1.  Factor  the  following  expressions : 

4a;2-12a;y  +  9y2;  x2  +  5cc  +  6;  x3-8f. 

2.  Find  the  G.C.D.  of 

2a;3  -  4a;2  -  13a;  -  7  and  6a?  -  11  x2  -  37 x  -  20.    • 

3.  Find  the  L.C.  M.  of  4(1  +  x),  4(1  -x),  and  2(1  -x2). 

4.  Simplify  ^  +  5_2a-5 6a&_. 

a  —  b        a-\-b       a2  —  b2 

m 

5.  Multiply  an  +  a2  and  Va  together. 

6.  Solve  the  equation  -^-  +  a  +  —  =  0. 

b  —  ex  c 

7.  Solve  the  simultaneous  equations 

5     "    10    ~U  anct6  +  ~T~~3' 

8.  Extract  the  square  root  of  x* — 2  x?y  +  3  ^y2  —  2  rzy3  +  y4. 

9.  Solve  the  quadratic  equation  x x~    =  — Hr — 

^  ^  8a? -3       a;  +  l 

10.    Solve  the  simultaneous  quadratic  equations 

-+-  =  5  and  i+~==13. 

x     y  xr      y2 

VI. 

Entrance  Examination,  Sept.,  1882. 

1.  Factor  the  following : 

9m*-24m+16;  x2  ~2xy  +  y2-  z2. 

2.  Find  the  G.C.D.  of 

12a;3  -  9  a;2  +  5x  +  2  and  24a;2  +  10a?  +  1. 

3.  Find  the  L.C.M.  of  x2  -  1 ;  a;2  +  2a;-3;  6a?2-a;-2. 


ALGEBRA.  81 


4a  ba2 


4.  Simplify  , 

r    J   x  —  a      \x  —  a)       {x  —  a) 

5.  Show  that 

(a+bV^l)(a-bV^)=(a^b+V2aJ)(a+b--y/2a~b). 

6.  Solve  the  equation  — ■  — - —  =  — 

ox  box 

7.  Solve  the  simultaneous  equations 

y  —  x      8  '  7 

8.  Extract  the  square  root  of 

^  +  3^  +  6^+ 7a;3 +  6^  +  3^+1. 

9.  Solve  the  quadratic  equations 

%  +  ~-f-  =  0a,ndl9xi+216x1  =  x. 
6       4       3a 

10.   Solve  the  simultaneous  quadratic  equations 

xv      ,         .,  a      b 

-  +  1  =  1,  and  -  +  -  =  4. 

a     o  x     y 

VII. 

Harvard  College,  Cambridge,  Mass. 

Examination  for  Admission,  June,  1883. 

1.  Solve  the  equation  1  =  2- 4aa'-3ft(*-2X 

*  x  2a(x2  +  l)  +  3b 

2.  A  man  walks,  at  a  regular  rate  of  speed,  on  a  road 

which  passes  over  a  certain  bridge,  distant  21  miles 
from  the  point  which  the  man  has  reached  at  noon. 
If  his  rate  of  speed  were  half  a  mile  per  hour 
greater  than  it  is,  the  time  at  which  he  crosses  the 
bridge  would  be  an  hour  earlier  than  it  is.  Find 
his  actual  rate  of  speed,  and  the  time  at  which  he 
crosses  the  bridge.     Explain  the  negative  answer. 


82  EXAMINATION    PAPERS. 

3.  Find  the  prime  factors  of  the  coefficient  of  the  6th  term 

of  the  19th  power  of  (a  —  b).     What  are  the  expo- 
nents in  the  same  term,  and  what  is  the  sign  ? 

4.  Eeduce  the  following  fraction  to  its  lowest  terms : 

X*-4:3?  +  10^-12^+9' 

5.  Prove  that,  if  a  :  b  =  c  :  d,       '     = .  =  -  =  -• 

c  -fa      c  —  d      c      d 

6.  Solve  the  equations,  xy  =  4  —  y2 ;  2x2  —  y2=17.     Find 

all  the  answers,  and  show  what  values  of  x  and  y 
belong  together. 

VIII. 

Yale  College,  New  Haven,  Ct. 

Examination  for  Admission,  Jy,ne,  1883. 
1.    Reduce  the  following  expression  to  its  simplest  form: 


x(x  —  a)(x  —  b)     a(a—x)(a—b)     b(b—x)(b  —  a) 

2.  Resolve  y*—b9  into  three  factors. 

3.  Change  xy~2  —  2#*y~12T1-f  z-1  to  an  expression  which 

will  contain  no  negative  exponents. 

^    lf  a  +  h  +  c  +  d  =  a-b  +  c-d  b    the      •  ci  les 

a+b  —  c  —  d     a  —  b  —  c  +  d 

of  proportion  that  -=  — 
b      d 

5.  Find  the  value  of  2 a  Vl  +  #2  when  x  =  -^(■\t  ~\~ )' 

6.  Given  (7-4V3)*2  +  (2  -  V3)rr  =  2,  to  find  x. 

7.  The  sum  of  two  numbers  is  16,  and  the  sum  of  their 

reciprocals  is  \.     What  are  the  numbers  ? 


ALGEBRA.  83 


8.    Compute  the  value  of  the  continued  fraction 
1  • 

2  +  -U 


*  +  l 


9.  Convert  — into  an  infinite  series  by  the  method 

of  indeterminate    coefficients,    or   by   the   binomial 
theorem. 

10.  Insert  three  geometrical  means  between  £  and  128. 


IX. 

Sheffield  Scientific  School,  New  Haven,  Ct. 

Entrance  Examination,  June,  1883. 

Candidates  for  examination  in  this  subject  as  a  whole  should 
take  the  whole  of  this  paper ;  those  for  the  first  year's  partial  exam- 
ination, the  first  part  of  it ;  those  for  the  second  year's  partial  exam- 
ination, the  second  part. 

State  what  text-book  you  have  studied,  and  to  what  extent. 

I. 

1.  Reduce  to  their  simplest  forms  the  fractions 

,.  x     ac-\-  bd+  ad-\-bc    ,     ,••  n  ax"1  —  bxm+1 
^  af+2bx  +  2ax  +  bf     W  a'bx-b'x3  ' 

2.  Given  — ^ — ^—l bx  =  ae  —  3  bx,  to  find  x. 

a  a 

3.  A  sum  of  money,  at  simple  interest,  amounted  in  m 

years  to  a  dollars,  and  in  n  years  to  b  dollars.    Find 
the  sum  and  the  rate  of  interest. 

4.  Prove  that  if <  1  —  -,  and  m  is  positive,  then  x<y> 

7Tb  X 


84  EXAMINATION   PAPERS. 

5.  (i.)  Simplify  (a2bs)l  +  (aV)i 

(ii.)  Extract  the  square,  root  of  6  hm2n  -f  h2  -f  9  m4n- 

(iii.)  Reduce  .  to  an  equivalent  fraction 

V#  +  a  —  V#  —  a 

with  a  rational  denominator. 

II. 

6.  Given  15^3  —  20^  =  35,  to  find  x. 

7.  Given  a7+^~9  _  (x  _  2)2,  to  find  ar. 

a?  —  V#2  —  9 

8.  Given  x2  —  xy  =  48  and  xy  —  y2  =  12,  to  find  x  and  y. 

9.  The  number  of  permutations  of  n  things  taken  r  to- 

gether is  equal  to  10  times  the  number  when  taken 
r  — 1  together;  and  the  number' of  combinations  of 
n  things  taken  r  together  is  to  the  number  when 
taken  r  —  1  together  as  5  to  3  ;  required  the  value 
of  n  and  r. 

3  4-  2x 
10.    Expand         "f    into  a  series  of  ascending  powers  of  x, 

by  the  method  of  indeterminate  coefficients.  (Four 
terms  of  the  series  will  be  sufficient.) 

X. 

Entrance  Examination,  Sept.,  1883. 
(State  what  text-book  you  have  studied,  and  to  what  extent.) 

1.  Given  x~~y^     =  a  and  x  ;  y~^    =  b  to  find  x  and  y. 

x  —  y  —  1  x-{-y  —  1  * 

2.  Simplify  (i.)  V27  +  2  Vi8  +  3  VlOa 

(ii.)  (V^b)\V^W)\ 
,...,  x2p(q-1)~y2q(p-1) 
(1U')    xp(q~1}  +  yq{2-l) 


ALGEBRA.  85 


Form  an  equation  whose  roots  shall  be  2  and  —  3.     Re- 
solve x2  —  3x-\-4.  into  two  factors. 

Given  -  4-  -  =  5  and  -=  +  - » =  13,  to  find  x  and  y. 
x  '  y  x2     y 


_    n.         3x+V4:X  —  x'2 

0.    Given .  =  2,  to  find  x.  ■% 

Sx-V^x-x2 

6.  To  deduce  a  formula  for  the  sum  of  a  geometric  pro- 

gression in  terms  of  the  first  term,  the  ratio,  and  the 
number  of  terms. 

7.  Having  10  different  letters,  how  many  sets  of  two  each 

can  you  form  of  them,  differing   by  at  least  one 
letter  ? 

8.  Expand into  a  series  of  ascending  powers 

1  —  Ax  -j-  x 
of  x  by  the  method   of  indeterminate  coefficients. 

(Four  terms  of  the  series  will  suffice.) 

9.  Express  log  \p— £-  in  a  form  adapted  to  computation. 

10.  To  deduce  a  formula  for  the  amount  of  a  given  sum 
of  money  for  a  given  time  at  a  given  rate  of  com- 
pound interest. 


XI. 

Amherst  College,  Amherst,  Mass. 

Examination  for  Admission,  June,  1883. 

1.  Find  the  value  of  6a-[4Z>-  {4a-(6a-45)}]. 

2.  Divide  a~Zn-  b6n  by  arn-  b2n. 

3.  Show  that  a0  =  1 ;    also,  that  a~m  =  1  -v-  am. 

4.  Resolve  a4m—  ¥m  into  its  prime  factors. 


86  EXAMINATION    PAPERS. 


5.  Find  the  G.C.D.  of  a4  -  V  and  a3  +  a2b  -  ab2  -  b\ 

6.  Given  ?>ax—2bx  —  \c  —  \mx=%c-\-\mx—ri—bx-\-2ax, 

to  find  x. 

7.  Divide  the  number  a  into  two  parts,  such  that  the  sec- 
*      ond  part  shall  equal  m  times  the  first  part  plus  n. 

8.'3y— 2a?  =  9;    7x  +  y  =  26  ;  find  x  and  y. 

9.    Multiply  Va  +  c  by  Va+~c. 

10.  Zx2 -4*  =  15.     Find*. 

11.  Expand  (1  -J-  #2)7  by  the  binomial  formula. 

12.  Find  the  (2n)  th  term  of  the  series  1,  3,  5,  7, 


XII. 

Williams  College,  Williamstown,  Mass. 

Entrance  Examination,  June,  1883. 

1.  Divide  *2  +  i  + 2  by  x  +  -- 

x2  J  x 

2.  Add  the  fractions and 


2a-2b  2b-2a 

3.    Simplify  1 


t+\ 


f 

4.  The  sum  of  two  numbers  is  5760,  and  their  difference  is 

equal  to  one-third  of  the  greater.    Find  the  numbers. 

5.  -+|  =  1;    ~-  +  £  =  !•     Findxandy. 
a      b  3a      66      6 

6.  Solve  the  equation  V3  x  -\-  4  -f  V3  ^  —  5  =  9. 


ALGEBEA.  87 


4.    Solve  3^-8;r~19-8=^^+5:r~88  +  10. 
2  4  3 


XIII. 

Tufts  College,  College  Hill,  Mass. 

Examination  for  Admission,  June,  1881. 

1.  Divide  2  am+l  —  2  an+1  —  am+n+  a2n  by  2  a  —  an. 

2.  Find  the  G.C.D.  of  ah  +  am,  6n  -f-  mn,  and  b2n  —  m2n. 

4        1 

3.  Amplify  _£ 

3a?- 

_   2 

5.  Solve  V4  -f  a;  =  4  —  V#. 

6.  Solve  I  +  ?  =  iland?  +  -  =  -- 

x      y      15         x      y      b 

7.  There  are  three  numbers  whose  sum  is  324 ;  the  second 

exceeds  the  first  as  much  as  the  third  exceeds  the 
second;  the  first  is  to  the  third  as  five  to  seven. 
What  are  the  numbers  ? 

x      h 

8.  -  +  -  =  c.     Find  the  values  of  x. 
a     x 

XIV. 

Trinity  College,  Hartford,  Ct. 

Examination  for  Admission,  June,  1883. 

(One  problem  may  be  omitted  in  each  of  the  three  divisions  indi- 
cated by  the  letters  A,  B,  C.) 

A. 

1.    Find  the  G.C.D.  of  2a;2  +  x-  1,  x2  +  5x  +  4,  anda^  +  l. 

o    Q1       .i             .•       6a;+7      2a;-2      2a;  — 1 
Z.    bolve  the  equation — = . 

H  15         7ar-6  5 


88  EXAMINATION   PAPERS. 

3.  Two  workmen  -together  finish  some  work  in  20  days ; 

but  if  the  first  had  worked  twice  as  fast  and  the 
second  half  as  fast,  they  would  have  done  it  in  15 
days.    How  long  would  it  take  each  alone  to  do  the 

work  ? 

B. 

4.  Multiply  2VZ^3-3V=2  by  4V^3+6V:r2;  di- 

vide V— 5  by  V— 1.  Explain  the  process  in  each 
case. 

5.  Solve  the  equation  V#—  3  —  V#  —  14  —  V4#  — 155  ==  0. 

Give  and  explain  the  rule  for  solving  a  quadratic 
equation. 

6.  Solve  the  equation  _^  + 1  =  _^L_  + -^  ; 

also,  x4  +  4*2  =  117. 

C. 

7.  Find  two  numbers  such  that  their  product  is  96,  and 

the  difference  of  their  cubes  is  to  the  cube  of  their 
difference  as  19  to  1. 

8.  In  an  arithmetical  progression,  a  =  3,  Z=42£,  d  =  2£) 

find  n  and  s.  Explain  the  rule  for  the  sum  of  a 
geometrical  progression. 

9.  Expand  {a  —  b)n  and  ( I  +  3  y  J  by  the  binomial  theorem. 

XV. 

Wesleyan  University,  Middletown,  Ct. 

Examination  for  Admission,  June,  1883. 

1.   When  a  =  1,  b  =  0,  c— %  find  the  value  of 

(Sa  +  2)(2a-\-b)-a\2c-a[(3a2-\-6)  +  c]\. 
Multiply  a*a* +  2ai3*  by  2tfx\-a\x\ 


ALGEBRA.  89 


2.  Factor  ^-64;    a2Z>2-3a& -4.  # 

3.  Glven  7g_l)-3(2y+8)  =  0;"2-^-™£±9  =  0; 

to  find  x  and  y. 

4.  Solve  at  least  two  of  the  following : 

(i.)  6a;2-13^  +  6  =  0; 

(ii.)  3^a/^+-^=16; 
(-)^  +  ^  =  8;   i  +  i=| 

How  do  you  "  complete  the  square  "  ? 

re- 


From  a-vl    z         take  &  a| — ,  and  express  the 


+ 
suit  in  its  simplest  form. 

6.  Write  the  repeating  decimal  0.3  as  the  sum  of  a  geomet- 
rical progression.     Find  the  limit  of  the  sum. 

XVI. 

Cornell  University,  Ithaca,  N.Y. 

Entrance  Examination,  June,  1882.  —  Elementary  Algebra. 

1.  Define :  known  and  unknown  quantities,  positive  and 
negative  quantities,  addition,  a  common  multiple  of 
two  or  more  numbers,  a  radical,  an  equation,  a 
theorem. 

i*  into  three  prime  factors. 


3.  Eeduce  the  fraction  ~j~   $ '  ^±JL  to   an   equivalent 

■Vx  —  y 
fraction  having  a  rational  denominator. 

4.  Divide  x  +  y  +  z  —  3  ~Vxyz  by  a£  +  y$  +  zi 


90  •         EXAMINATION    PAPERS. 

5.  For  $8  1  can  buy  2  pounds  of  tea,  10  pounds  of  coffee, 

and  20  pounds  of  sugar,  or  3  pounds  of  tea,  5  pounds 
of  coffee,  and  30  pounds  of  sugar,  or  5  pounds  of  tea, 
5  pounds  of  coffee,  and  10  pounds  of  sugar.  What 
are  the  prices  ? 

r>    a  i      x-u  o,x  —  b  ,  a      bx      bx  —  a 

6.  Solve  the  equation   ■ f-  -  = — . 

H  4  3       2  3 


7.  Solve  the  equation  x  -f  5  +  V^+5  =  6,   giving  all  the 

roots. 

8.  Solve  the  equation    x  +  a  +x~2a=l,  and  get  the 

^  x—2a       x+a  8 

sum  and  the  product  of  the  two  roots. 

XVII. 

Entrance  Examination,  June,  1882.  —  Advanced  Algebra. 

1.  Prove  the  formula  for  the  development  of  (a  +  x)n ;  and 

from  this  formula  get  the  development  of  (l-f-'tf-f-rc2)3, 
and  four  terms  of  (a2  —  x2)~%. 

2.  Prove  the  formula  for  the  sum  of  a  geometrical  progres- 

sion, the  first  term,  the  ratio,  and  the  number  of 
terms  being  given.  From  this  formula  obtain  an 
expression  for  the  amount  of  a  deferred  annuity  at 
compound  interest ;  the  annual  payment,  rate,  and 
time  being  given. 

3.  By  the  method  of  differences,  find  log  24,  by  continuing 

the  series :  log  20  =  1.3010,  log  21  =  1.3222, 
log  22  =  1.3424,  log  23-  1.3617. 
From  the  same  data  find  log21J'by  interpolation. 

4.  Prove  that  loga5  X  log6#  =  loga#. 

5.  By  the  method  of  undetermined  coefficients  prove  that,  if 

y  =  x  +  x2  -f  x3  -}- ,  then  also  <c  =  y  —  y2  +  y3— 

6.  By  continued  fractions,  find  five  successive  approxima- 

tions to  the  value  of  V2. 


ALGEBRA.  91 


7.  Depress  the  equation  x*-\-  2x3  +  x2  =  35  a;  +  74,  by  re- 

moving a  commensurable  root,  and  then  find  an  in- 
commensurable root  correct  to  two  decimal  places. 

8.  Prove  that,  if  the  coefficients .  of  an  equation  with  one 

unknown    quantity   be   real,    any   imaginary    roots 
enter  it  in  pairs. 

XVIII. 

Stevens  Institute  of  Technology,  Hoboken,  N.J. 

Specimen  Entrance  Examination  Paper. 

1.  Multiply  V2a  by  -VSa. 

2.  Divide  1  —  x  by  l-\-  x  to  four  terms. 

3.  Divide  V3a  by  V2o?. 

4.  Solve  l--gg-ng  =  0andq-2y-3==l. 


by  —c  1 

Sx  —  2y 

5.    Solves—  — =  2:    4x  —  dy  =  2. 

oi/  —  x 


6.  Solve  -\/a  —  x  —  Vo  —  x  =  -y/c  —  x. 

7.  Solve  Zxy  —  by2  =  2;     bxy  +  Sx1  =  l. 

8.  Solve  * -  =  a;    bx — *—= — -  =  ay. 

y     x  5  *  • 

9.  Establish  the  equation  for  the  permutations  of  n  things 

taken  r  at  a  time. 

10.  Prove  that  a  proportion  taken  by  division  is  a  true 

proportion. 

11.  If  x  varies  as  a-\-by,  and  when  x  =  l,   y  =  2;     and 

x  =  2,  y  =  —  5;  show  that  7x  =  9  —  y. 

12.  In  an  arithmetical  series,  given  the  common  difference, 

first  term,  and  number  of  terms,  to  find  the  sum  and 
the  last  term. 

13.  In   a   geometrical   progression,   given   the   first  term, 

number  of  terms,  and  the  last  term,  to  find  the  sum 
of  the  terms. 


92  EXAMINATION   PAPERS. 

XIX. 

Madison  University,  Hamilton,  N.Y. 
Entrance  Examination,  June,  1882. 

1.  Define :  (i.)  positive  and  negative  quantities ;   (ii.)  sys- 

tem of  notation  ;  (iii.)  similar  terms  ;  (iv.)  ratio  ; 
(v.)  compound  ratio  ;  (vi.)  proportion. 

m         t  m  t  m  t 

2.  Multiply  a*h~~*  —  as*&~s  +  l  by  aFlb~r°  +  l. 

3.  Eationalize  the  denominators  of  the  following  fractions  : 

1       aT  3 

(2f    £    (3)H(5)§ 

4.  Given  — ==       ~j~  — r~,  to  find  the  value  of  x. 

2--M  40  + a* 

5.  Four  given  numbers  are  represented  by  a,  b,  c,  and  d ; 

what  quantity  added  to  each  will  make  them  pro- 
portional ? 

6.  Suppose  a  body  to  move  eternally  in  this  manner:  viz., 

20  miles  the  first  minute,  19  miles  the  second  min- 
ute, 18^-  miles  the  third  minute,  and  so  on  in  geo- 
metrical progression  ;  what  is  the  utmost  distance  it 
can.reach  ? 

7.  A  hare,  50  of  her  leaps  before  a  greyhound,  takes  4  leaps 

to  the  greyhound's  3 ;  but  2  of  the  greyhound's 
leaps  are  equal  to  3  of  the  hare's.  How  many  leaps 
must  the  greyhound  take  to  catch  the  hare  ? 

XX. 

Vassar  College,  Poughkeepsie,  N.Y. 

Specimen  Examination  Paper  for  Admission. 

1.    Factor  the  following  expressions : 

9#*  —  y2 ;  ra2  —  2ran +  n2 ;  ns  —  n ;  mi—ni;  x3  +  as. 


ALGEBRA.  93 

2.  What  is  the  rule  for  transposing  a  term  from  one  side 

of  an  equation  to  the  other ;  what  is  the  principle  ? 

3.  Solve  the  equation 


a     a-\-b      a  —  b 

27a~3b20 

4.  Find  the  fifth  power  of  2  a2 ;  the  fourth  root  of  • 

lodur 

5.  Square  x ;  cube  3  -f-  V2". 

6.  A  and  B  engage  to  mow  a  field.      A  alone  can  mow  it 

in  b  days,  and  B  alone  in  c  days.     In  what  time  can 
both  together  mow  it  ? 

7.  Solve  the  equation  V#  +  4  —  V#  =  V#  +  f . 

8.  Solve  the  the  equations 

aj»-^  =  2J5;  ^2  +  ^  +  y2  =  43. 

9.  Find  four  values  of  x  in  the  equation  x2  +  —  =  a2  -f  —2- 

10.  Form  the  equations  whose  roots  are  a,  —a,  and  b. 

11.  Deduce  the    formula  for   the   sum   of  n   terms   of  a 

geometrical  progression. 


XXI. 

College  of  New  Jersey,  Princeton,  N.J. 

Examination  for  Admission,  Sept.,  1883. 

1.  What  text-book  have  you  used  ? 

2.  Simplify  the  expression  i 5 Ja*  +  P 


b 


a 


94  EXAMINATION    PAPERS. 

3.  Solve  the  following  equations : 

5-6g  I   ^  +  14-17-3:r     4^  +  2 
3  5  3      ' 

2U+V*/        *-2        2*        3' 
^  +  a^  =  6. 

4.  Solve  the  simultaneous  equations  : 

""Iff.      V    — i —     a. 

5.  Divide  x-^-.y  +  z  —  Z&ytzt  by  a£ -f- yi -}- zi. 

6.  Find  the  square  root  of 

16a;4  -  16a^2  +  ISVx1  +  4a2  62  -  8a63  +  464. 


XXII. 

John  C.  Green  School  of  Science,  Princeton,  N.J. 

Entrance  Examination,  Sept.,  1883. 

1.  What  text-book  have  you  used,  and  how  much  of  it  have 

you  studied  ? 

2.  Solve  the  following  equations : 

(3a  —  x)(a  +  2x)  =  (5a  +  x)(a  —  2x); 

l-^zl-*Lz|;     (*»-5)a  +  29(^-5)  =  96. 

3.  Solve  the  following  pairs  of  simultaneous  equations : 

^-y-31;      57+^  =  33; 
h       J  '  5 

3»  +  ^2  =  85  ;     ay  =  42. 

4.  Form  the  quadratic  equation  whose  roots  are  -|  and  -§. 

5.  Add  v%    \^,  and  \f. 

6.  Find  the  square  root  of#4+— -  +  4a£— a?  +  4a£— 2 a:-*. 


ALGEBRA..  95 

XXIII. 

University  op  Pennsylvania,  Philadelphia,  Pa. 

Examination  for  Admission,  June,  1882. 

1.  Divide  a;6-  9a:5 -f  25a;4-  Sx3-  98  a;2  +  156  a;  -  72  by 

x3  —  2a?  —  Ax -{-8  by  synthetic  division. 

2.  Simplify  - — — ^ . +  — -^ -  +  - — rr- 

(a— b)(a— c)      (b  —  a)(b  —  c)      (c—a)(c—b) 

3.  Simplify 

•      2^1^-^i7-2^^+2^4¥8-2^+3^875. 

4.  Find  the  difference  between  2  + 1  -f  I  + to  infinity, 

and  i  +  £  +  I  + to  12  terms. 

5.  Find  the  square  root  of 

a;8-12^+62^6-180^4-321^-360^+248^-96^+16. 

6.  Find  the  value  of  x  from  V2aT+6  +  V2a;-5  =  11. 

7.  Find  x  and  y  from 

x      ,      3  7        ,      2  3  1 

and 


a;  +  2     2/-3      12  a;  +  2     y-3  12 

8.  Solve  the  equations  x-\-y  —  z  =  4,  2a;—  3y-f  4z  =  20, 

and  5  a;  —  6y  +  3  z  =  10. 

9.  Find  a;  from \ =  1+1+1. 

a-\-  b  +  x     a     o      x 

10.   Solve  the  equations  #*+  10  xy  =  11  and  5xy  —  2>y2  =  2. 


XXIV. 

University  of  Michigan,  Ann  Arbor,  Mich. 

Examination  for  Admission,  June,  1883. 

Define  exponent.  Illustrate  the  significance  of  fractional 
and  negative  exponents.  Write  ^/ax2y~m  without 
using  the  radical  sign. 


96  EXAMINATION    PAPERS. 

2.  Define  elimination.     What  method  do  you  prefer,  and 

why  ?     Apply  to  the  equations 
a  ,  b  c  ,  d 

x      y  x     y 

3.  Expand  (a2  +  x2)%  by  the  binomial  formula. 

4.  Solve  the  equation   3V5-4=  15  + VSs 

2  +  Va?       40  +  Val 

5.  Produce  the  formulae  for  last  term  and  swra  in  arith- 

metical and  geometrical  progression. 

6.  Simplify   ^ EL,    * fLL.. 

r    J  a;  a;2 

7.  Rationalize  the  denominator  of 

•vV  +  a?+r+  yV  +  a:  — 1 
V^2  +  ^  +  l- V^  +  a;-l 

8.  Solve  the  simultaneous  equations 

4(^  +  2/)  =  3o;y;    #  +  y  +  a;2  +  y2  =  26. 

9.  Solve  the  simultaneous  equations 

a;2  +  n*y  +  y1  =  52  ;    #?/  —  x2  =  8. 
10.    Define  logarithm,  mantissa,  characteristic.      How  can 
you  extract  roots  by  logarithms  ? 
Given  log  x  =  2.301030,  what  is  loga;i? 


XXV. 

Lake  Forest  University,  Lake  Forest,  III. 

Entrance  Examination,  June,  1882. 

1.  Define  coefficient,  exponent,  similar  quantities,  monomial, 

binomial,  equation. 

2.  Find  the  G.C.D.  of 

a;4  +  2a;2  +  9and  7xz-  11a*  +  15a?  +  9. 


ALGEBRA.  97 

3.  Find  the  value  of  x  in 

Sx  —  1  .  5  —  x     2^— -4__g ff-f 2 

7^4  12     ~  28   ' 

4.  Explain  the  rule  for   subtraction,   showing   why   the 

signs  of  the  subtrahend  are  changed.     Illustrate  by 
diagram  or  numbers. 

5.  A  man  rows  a  boat  with  the  tide  8  miles  in  48  minutes, 

and  returns  against  a  tide  two-fifths  as  strong  in  80 
minutes,  what  is  the  rate  of  the  stronger  tide? 

6.  The  product  of  two  numbers  is  702,  and  their  sum  is  60. 

Find  the  numbers. 

7.  Factor  x*-2x-3  and  ^ -^- 13a;  +  24. 


0    a  ,       -y/a—x      Va  — #        ,- 

o.    bolve =  -y/x. 

x  a 


9.   Solve  a; -f  5V37-z=:43. 
10.    What  number  added  to  its  reciprocal  makes  2.9? 

XXVI. 

Education  Department,  Ontario. 

Examination  of  Third  Class  Teachers,  July,  1883. 

1.  Divide 

(i.)  (a-b)c3  +  (b-c)a*  +  (c-a)bz  by 
(a  —  b)(b  —  c)  (c  —  a)) 

^  £±£_£±£  h    i_i. 

x3^  x2^  x     y 

2.  "What  must  be  the  values  of  a,  b,  and  c,  that  x3  -f-  ax2 

-\-bx-\- c  may  have  x—1,  x—2,  and  x  —  3  all  as 
factors  ? 

3.  Find  the  H.C.F.  of 

(i.)  3x*-4:X3  +  l{m&4:X*-5xi-x2  +  x  +  l; 
(ii.)  8xs-if+27z3+l8x!/z  and  4o*+12a?2+ 9z2-y\ 


98  EXAMINATION    PAPERS. 

4.  Simplify 

(n/4a»     -A/    2x       {\,(Z*     {\f    4s»+2sy        1\, 
WVy2        A2^~y      /     W        A^+^y+y2       /' 

rin3»+(q  +  6)3»  +  (a6  +  l)tt+6 

k   '}  bx^  +  (ab-{-V)^+{a  +  b)x+l 

5.  Find  the  value  of  x  that  will  make  <^  +  ^+ad+bc 

independent  of  c  and  d. 

6.  (i.)  If  «  +  6  +  c  =  0,  then  -I  +  l+-L=  ji+l+I}2. 

a2      o2      c2       C  a     6     c ) 
(ii.)  If  x  =  a2  +  b2  -f  c2  and  y  =  «6  -f-  5c  +  ra, 

then  rr3  +  2tf  -  Sxf  =  (a3  +  63  +  c3  -  Zabc)2. 
(iii.)  If  2a  =  y  -j-z,  25  =  2;  + a:,  2c  =  .T  +  y,  express 
(a  +  5  +  c)3  -  2  (a  +  5  +  c)  (a2  +  62  +  c2) 
in  terms  of  or,  y,  and  2;. 

7.  Find  a  value  of  a  which  will  make  the  quantities 

(a+b)(a  +  c)    and    (a  +  c)(a  +  d) 

a-\-b  -{-  c  a-\-  c-\-  d 

equal  to  one  another. 

8.  Solve  the  equations 

(i.)  V^r  +  3  +  Vx  +  2  =  5; 

,..  s  b—x  .  b  —  2x  .  ar-f-1      2-\-5x     n 

(11. ) ! ! =  0  : 

V    ;      3     ^      4      ^     3  2 

(iii.)  O  +  a  +  $)(<?  +  <*)  =  <>  +c  +  d)(a ■+  5), 

where  c?  +  c?  is  not  equal  to  a  -f  &. 

9.  One  side  of  a  right-angled  triangle  exceeds  the  other 

by  3  feet,  neither  being  the  hypotenuse,  and  its  area 
is  18  square  feet.  What  are  the  sides  ? 
10.  A  cistern  with  vertical  sides  is  h  feet  deep.  Water  is 
carried  away  from  it  by  one  pipe  |-  as  fast  as  it  is 
supplied  by  another.  Find  at  what  point  in  the 
side  the  former  pipe  must  be  inserted  that  the  cis- 
tern may  fill  in  twice  the  time  it  would  did  water 
not  flow  from  it  at  all. 


ALGEBRA.  99 


XXVII. 

University  of  Toronto,  Toronto,  Ont. 

Junior  Matriculation.  — Annual  Examination  for  Honors  in  Algebra, 

1883. 

1.  Find  the  product  of  (a  +  b),  (a2  +  oh  +  b2),  (a  -  b), 

and  (a2  —  ab-\-  b2). 

2.  If  a  and  b  are  positive  integers,  show  that 

xa  x  xh  =  xa+h. 

3.  Prove  the  rule  for  finding  the  G.C.M.  of  two  quantities. 
Find  the  G.C.M.  of  6^+  Wx4y-  4^V-  lO^yz2  and 

9x3y  -  27 xlyz  —  6xyz2  +  18yz3. 

4.  State  the  rule  for  extracting  the  square  root  of  a  com- 

pound quantity. 
Extract  the  square  root  of  x2  -f-  y. 

5.  Solve  the  following  equations  : 

(i.)  3a?-M  =  li;  2y  +  3z  =  16,  5>  +  4y  =  35; 


(ii.) 


x  -\-  a      x  -\-b 
x  —  a      x— b 


(in.)  _  +  _=2+- 
ax  x 

6.  When  are  quantities  said  to  be  in  geometrical  progres- 
sion, when  in  harmonical  progression,  and  when  in 
arithmetical  progression  ?  (i.)  Find  two  harmonical 
means  between  a  and  b.  (ii.)  The  first  term  of  a 
geometric  series  is  •£■,  the  ratio  J,  and  the  number  of 
terms  is  6  ;  find  the  sum  of  the  series. 


100  EXAMINATION    PAPERS. 

7.  Show  that  the  number  of  combinations  of  n  things  taken 

r  together  is 

n(n—I)(n  —  2) (n  -  r  -f  1) 

1x2x3 r 

How  many  words  of  four  letters  can  be  formed  out  of 
the  first  13  letters  of  the  alphabet,  having  one  vowel 
in  each  word  ? 

8.  Expand  to  five  terms  (a  -f  5)~*. 
Show  that   (ji|Y 

-l  +  ^  +  |(^2  +  ^)  +  |(^  +  ^)  +  T^(^+^)  + 

9.  A  number  consists  of  two  digits :  when  the  number  is 

divided  by  their  sum  the  quotient  is  4,  and  when 
divided  by  their  difference  the  quotient  is  12';  find 
the  number. 
10.  The  crew  of  a  boat  rowed  six  miles  down  a  river,  and 
half-way  back  again,  in  2  hours.  Supposing  the 
stream  to  have  a  current  2\  miles  an  hour,  find  at 
what  rate  they  would  row  in  still  water. 

XXVIII. 

College  of  Ottawa,  Ottawa,  Can. 

Matriculation  Examination.    Session  1882-83. 

1.  Translate  the  following  into  common  language : 

l  +  2a  ,+  * 

2.  Divide  (i.)  2a2 b  +  bd  +  a3  -f  2ab2  by  a2  +  b2  +  ab ; 

(ii.)  12w~2y~4  by  —4 xy2. 

3.  Find  the  prime  factors  of  25<?2c?4  —  9a6c4. 

Find  the  G.C.D.  and  L.C.M.  of  9mx2—6mx  +  m  and 
9  nx%  —  n. 


ALGEBRA.  J.01 


4.    Find  the  sum  of 


a-\-  b           ,          b-\-  c  , 

~r  7 rz 7T  ~r 


(b  —  c)(c  —  a)      (c  —  a)  (a—b)      (a  —  b)(b  —  c) 
Find  the  algebraic  sum  of 

__3 7  4-20a; 

\-2x     2x+l      4a*- 1 

r-    a  ,  3a;  — 3  ,  A      20-x      6a;  —  8  ,  4a;-4 

5,  Solve, 5~  +  ^—2 -^-  +  -^- 

Find  a;,  y,  and 'z  in  the  following  equations : 

#  +  2/  +  ^  =  24  ;    x  —  ?/  -f  z  =  8 ;    a;  +  3/  —  z  =  6. 

6.  Raise  — — - — %-  to  the  cube. 


7.  Give  the  simplest  form  of  "\3V3. 

8.  Show  why  the  square  may  be  completed  in  the  quad- 

ratic 3  a;2  —  7  a;  =  20,  by  the  same  rule  as  in  x2  +  2x 
=  24,  without  introducing  fractions. 

9.  Given  x2 -  2xy  =  12  and  x2-y2  =  12,  to  find  x  and  y. 

10.  A  boy  being  asked  how  many  sheep  his  father  had, 
replied  that  f  of  \  the  flock  would  be  25  less  than 
the  whole  flock.     How  many  sheep  had  his  father  ? 


XXIX. 

College  of  Ottawa,  Ottawa,  Can. 

Matriculation  Examination,     Session  1883-84. 

1.   Clear  away  the  parentheses,  and  reduce  the  following 
expression : 

a  +  b-(2a-Sb)~4:(5a  +  7b)~(-lSa  +  2b) 
+  S{a-e>(b-a)l 


102  EXAMINATION   PAPERS. 

2.  Give  the  three  formulas  for  the  expansion  of  (a  -f-  b)2, 

(a  —  b)2,  and  (a  +  b)(a  —  b),  and  give  an  example 
for  each  formula. 

3.  Divide  5a?-  3  -  4  a;2  +  #4  +  #3  by  -3  +  x2-2x. 

4.  Find  the  G.C.D.  and  the  L.C.M.  of  the  three  following 

expressions  : 

(2ar-4)(3a?-6);  (s-3)(4a;-8).;  (2a?-6)(5a?-10). 

wi2  +  w2      0 

.    Simplify    , , X 

n     m 

6.  Solve  the  equations 

2x+4y-Sz  =  22;  4:X-2y+5z  =  18 ;  5x+7y-z  =  63. 

7.  Extract  the  square  root  of 

15a4b2  +  a6  -  6  a5  5  -  20a3 58  +  b«  +  15a2  64  -  6aZ>5. 

8.  Convert  V I  into  such  an  expression,  not  a  decimal,  as 

shall  not  necessitate  two  extractions  in  finding  the 
cube  root  of  f . 

9.  Solve  the  following  equation :   I  x2  —  lx  +  20 1  =  421 . 

10.  The  hypotenuse  of  a  right-angled  triangle  is  20  feet, 

and  the  area  of  the  triangle  is  96  square  feet.     Find 
the  length  of  the  legs. 

11.  Find  the  tenth  term,  and  the  sum  of  ten  terms,  of  the 

series  1,  4,  10,  20,  35. 

12.  Develop — into  an  infinite  series  by  the  method 

of  undetermined  coefficients. 

13.  Find  the  value  of  x  in  the  equation  5X  ==  30. 


ALGEBRA.  103 

XXX. 

McGill  University,  Montreal,  Can. 

School  Examination,  June,  1883. 

1.  Multiply  1  +  2a?  —  a?—  ?#3  by  itself,  and  find  the  value 

of  the  result  if  1  —  2x  =  3. 

2.  Find  the  remainder  when  a5-4a362-8a263-17a&4-1565 

is  divided  by  a2  —  2  ab  —  3  b2. 

3.  Simplify  ix(x+V)\x+2-l(2x+l)\'t   Zg^g+l. 

4.  Reduce  the  following  fractions  to  their  lowest  terms : 

a2x-{-a3  (  _      (#*— q4)(ff— a)         t  1-f-^3 

ax2- a3''  {x2+a2-2ax){ax+x2Y  l+2x+2x,+a? 

5.  Find  the  square  root  of 

^4+2^3-^  +  landof  4^-4a?  +  l. 
4  9x2  +  6a;  +  l 

6.  Solve  the  equations 

(i.)  2x  —  -  =  18 ;  (ii.)  (m  +  w) (m  —  x)  =  m (n—x) ; 
(iii.)  2^-^  =  4;    3y  +  ^  =  9. 

7.  If  aa^  -\-bx-\-c  becomes  8,  22,  42,  respectively,  when  x 

becomes  2,  3,  4,  what  will  it  become  when  x  =  —  \  ? 

8.  Find  two  numbers  which  produce  the  same  result,  7, 

whether  one  be  subtracted  from  the  other,  or  the 
latter  be  divided  by  the  former. 

9.  In  a  certain  school  there  are  6  boys  to  every  5  girls ;  if 

there  were  2  boys  less  and  2  girls  more,  there  would 
be  the  same  number  of  each.     Find  the  number. 

10.  Any  odd  number  may  be  represented  by  2v+l.  Prove 
that  the  difference  of  the  squares  of  any  two  odd 
numbers  is  exactly  divisible  by  8. 


104  EXAMINATION    PAPERS. 

XXXI. 

University  of  Cambridge,  Eng. 

Second  General  Examination  for  the  Ordinary  B.A.  Degree,  Nov.,  1880. 
Time  allowed,  3  hours. 

, I     a1      ,,  ,.        r,         x-2      a;  +  23      10  +  a; 

1.  Solve  the  equations  (i.)  x -— —  =  — ~ — ; 

o  4  5 

(ii.)-2_=-$-_;    (iii.)  1  +  1  =  43,   2  +  |  =  42. 

x—b      x—a  9      8  8      9 

2.  Solve  the  quadratic  ax2  +  &£  +  c  =  0,   and  determine 

the  condition  that  its  roots  may  be  equal. 
If  a,*  ft  be  the  roots,  form  an  equation  whose  roots  are 

-  and  -• 
a  jS 

3.  Solve  the  equations 

r  v  Sx  +  2  ,  3a;  — 2      4a;2  +  12a;  +  2 

(l.)  ! ■ ! —  ; 

k  ;    x-3        x+S  x2-9 

(ii.)  Vx  +  Va  +  x  —  —z  I 

(iii.)  (*  +  2y)  (2a;  +  y)  =  20,  4ar  (a;  +  y)  =  16  -  ?/2. 

4.  The  first  term  of  an  arithmetical  progression  of  n  terms 

is  a,  and  the  last  term  I.     Find  the  sum,  and  also 
the  common  difference. 
If  n  be  odd,  and  the  sum  of  the  even  terms  be  sub- 
tracted from  the  sum  of  the  odd,  show  that  the 

result  is  • 

2 

5.  Find  the  sum  of  n  terms  of  a  series  in  geometrical  pro- 

gression. 
If  the  sum  of  a  geometrical  series  to  infinity  be  n  times 

the  first  term,  show  that  the  ratio  is  1 . 

n 

6.  Find  the  sum  of  the  series : 

(i.)  2    +2£  +  2*  + ,  to  12  terms; 

(ii.)     i—    I  +    f  — ,  to  8  terms; 

(iii.)  31  +  21  +  H  + ,  to  infinity. 


ALGEBRA.  105 


7.  Show  that  a  ratio  of  greater  inequality  is  increased  by 

taking  the  same  quantity  from  both  its  terms. 
Show  that  the  ratio  a  —  x  :  a  -j-  x  is  greater  or  less  than 
the  ratio  a2  —  xz :  a2  +  x2,  according  as  the  ratio  a  :  x 
is  one  of  less  or  greater  inequality. 

8.  Define  proportion.     When  are  quantities  said  to  be  in 

continued  proportion  ? 
If  a,  b,  c,  d  be  in  continued  proportion,  show  that 
fa -bV__  a 
[b  -  c)      d 

9.  When  is  one  quantity  said  to  vary  directly  and  when 

inversely  as  another? 
The  volume  of  a  sphere  varies  as  the  cube  of  its  radius : 
if  three  spheres  of  radii  9,  12,  15  inches  be  melted 
and  formed  into  a  single  sphere,  find  its  radius. 
.10.  A  and  B  start  simultaneously  from  two  towns  to  meet 
one  another.  A  travels  2  miles  per  hour  faster 
than  B,  and  they  meet  in  7  hours ;  if  B  had  trav- 
elled 1  mile  per  hour  faster,  and  A  at  only  half  his 
previous  pace,  they  would  have  met  in  9  hours. 
Find  the  distance  between  the  towns. 

11.  A  wine-merchant  buys  spirit,  and  after  mixing  water 

with  it,  sells  the  mixture  at  two  shillings  per  gallon 
more  than  he  paid  for  the  spirit,  making  23 1  per 
cent  on  his  outlay :  if  he  had  used  double  the  quan- 
tity of  water  he  would  have  made  37 ?  per  cent ; 
what  proportion  of  water  was  there  in  the  mixture  ? 

12.  Two  elevens,  A  and  B,  play  a  cricket  match.     A's  first 

innings  is  the  square  of  the  difference  of  B's  two 
innings,  and  A's  second  one-third  the  sum  of  B's 
two  innings ;  A  scored  60  more  their  first  innings 
than  in  their  second,  and  lost  the  match  by  one  run. 
What  were  the  respective  scores,  B  having  first 
innings  ? 


106  EXAMINATION   PAPEES. 

XXXII. 

University  of  Cambridge,  Eng. 

Second  Previous  Examination,  Dec,  1880. — Time  allowed,  2£  hours. 

1.  Define  coefficient,  term. 

Find  the  coefficient  of  x  in  the  expression 

^_|2a-6(C-*)j. 

2.  Find  the  continued  product  of 

x2  +  3x  +  2,  x2  —  5#  +  6,  ^  +  2^  —  3, 

and  multiply  together 

^  +  (V2-l)o;  +  l,  *2-(V2  +  l>  +  l. 

3.  Divide   x*-(b-2)xi-(2b-l)x'i-(b2+2b-8)x+3b+Z 

by  af  +  Zx  +  b  +  l. 

4.  Simplify    (i.)  -X--J 


(ii.) 


x  —  2  x'2  —  3#  +  2     r5  — 4#  +  3 
l+x      1+x2 
1  +  x2      1  +  a3 
1  +  xl      1  +  x3' 


1  +  a8     1  +  ** 

If  a  measures  both  6  and  c,  prove  that  it  will  measure 

the  sum  of  any  multiples  of  b  and  c. 
Find  the  G.C.M.  of  1  +  x  +  xz-  x>  and  1  -  **-  xe+  x\ 
Solve  the  equations 

—  5     x 

I 

1 


,  3x  —  5     #  +  l_o 


(lL)  2<>+3)~30  +  2)  '  60+1)' 
(m.)i(x  +  i/\=i(x-y\   3z+lly  =  4; 

(iv.)  3*2  +  l  =  ?~; 

f    , x+a .  b      x2  +  ab  . 

(v.)  — -7-7  +  -= ' 

a  +  6      a         ax 

(vi.)  0+1)  (y +  2)  =  10,   zy  =  3. 


ALGEBRA.  107 


7.  If  a  +  b  =  1,  prove  that  (a2  -  b2)2  =  a3  +  b*  -  ab. 

8.  If  j  =  -,  prove  that  each  of  these  fractions  is  equal  to 

b      a 
a-\-  one 
b-\-md 
If  a  +  b,  b-\-c,  c-\-a  are  in  continued  proportion,  prove 
that  b  -\-  c,  c  -\-  a,  c  —  a,  a  —  b  are  proportionals. 

9.  When  is  one  quantity  said  to  vary  as  another  ? 

If  -  +  -  varies  inversely  as  x  +  y,  prove  that  x2  +  y2 
x     y 

varies  as  xy. 


XXXIII. 

University  of  Cambridge,  Eng. 

General  Examination  for  the  Ordinary  B.A.  Degree,  June,  1881. 
Time  allowed,  3  hours. 

1.  Prove  that  (x  +  4)3-  (x  +  l)3  =  9  (x  +  1)  (x  +  4)  +  27. 

2.  Simplify 

(l0  (o  +  a+c)A+i+iW*  +  «0(« +  *)(«  +  »). 

w  v     '  \ax  bej  abc 


r.  x  a262 


62  -  a2  -  62  +  1 


a&  —  a  —  b  -f- 1 

3.  Find  the  G.C.M.  of  ^-4^2+2a;+3  and  2xi-x2-5x-3. 

4.  Prove  that,  if  on  and  n  be  positive  integers, 

(ar)n  =  (an)"\ 
Prove  that  (V3)3V3  -  (3  V3)V3. 
6.    Solve  the  equations 

(i.)  i(5*  +  l)  +  ^3  =  z+J|; 

(n.)-i-  +  rJ-  =  ?L±l. 

(iii.)  ax  -f  6y  =  2,    ab  (x  +  y)  =  a  +  b. 


108  EXAMINATION    PAPERS. 

6.  Show  that  the  product  of  the  roots  of  the  equation 

ax2  +  bx  -f-  c  =  0  is  — 
a 

Prove  that  the  difference  of  the  roots  of  the  equation 
x2  +  px  +  q  =  0  is  equal  to  the  difference  of  the 
roots  of  the  equation  x2  +  Spx  +  2p2  -f  q  —  0. 

7.  Solve  the  equations 

(i.)  (x-2)2  +  (x  +  b)2  =  (x+7)2; 
(ii.)  ax2  -{-  2bx  =  a  —  2b  ; 
(iii.)  V#  +  2/  =  Vy  +  2,   x  —  y  =  7. 

8.  Find  the  sum  of  w  terms  of  a  G.P.  of  which  the  first 

and  second  terms  are  a  and  b. 
If  each  term  of  a  G.P.  be  squared,  prove  that  the  new 
series  will  also  form  a  G.P. 

9.  If  f  =  4  prove  that  each  of  these  fractions  is  equal  to 

pa  +  qc 
pb-\-qd 

10.   What  is  meant  by  saying  that  A  varies  as  B  ? 

If  the  volume  of  a  cone  varies  jointly  as  its  height, 
and  the  square  of  the  radius  of  its  base,  show  that 
if  the  heights  of  three  cones  of  equal  volume  are  in 
continued  proportion,  so  also  are  the  radii  of  their 


11.  Find  a  fraction  such  that  the  denominator  exceeds  the 

square  of  half  the  numerator  by  unity,  and  the  prod- 
uct of  the  sum  and  difference  of  the  numerator 
and  denominator  is  64. 

12.  A  vessel  is  half  full  of  a  mixture  of  wine  and  water. 

If  filled  up  with  water,  the  quantity  of  water  bears 
to  that  of  wine  a  ratio  nine  times  what  it  would  be 
were  the  vessel  filled  up  with  wine.  Determine  the 
original  quantities  of  wine  and  water. 


ALGEBRA.  109 


XXXIV. 

University  of  Cambridge,  Eng. 

Previous  Examination,  June,  1881.  —  Time  allowed,  2\  hours. 
Elementary  Algebra. 

1.  Simplify  0  -  of  -  (x  -  b)2  -  (a  -  b)  (a  +  b  —  2>x)  ; 

and  find  the  value  of 

S(a-b)(a  +  2b)  +  2(a-2b)(2a-b)  +  2(2a-by, 

when  a  =  0  and  b  =  —  2. 

2.  Divide  7a3- 22a26 +  4a62- 363  by  a  -36. 

3.  Resolve  into  factors 

a;2-2a;-255;  21a^-13a;y-20y2;  (o;  +  2y)3-y3. 

4.  Simplify 

(i)  a2- 16a;— 17. 
lLj  a;2-  22a;  +  85' 

5.  Find  the  L.C.M.  of 

(a*  -  tf),  (2x2  -  Zxy  ;  +  y2),  and  (a;3  +  x2y  +  xy>). 

6.  Solve  the  equations 

00  ~2~  +  ~^ 5~  +  5' 

(ii.)  2a?(7a?-10)  =  13(a?-l); 

(iii.)  7  +  ^|+  4  =  0; 

x  —  6      (x—  l)(x— 3) 

(iv.)  5abx  +  2y=ie>b,    3abx  +  4:y=18b  ; 
(v.)  a;2  +  3a;y  =  - 8,   y2-xy  =12. 

7.  Extract  the  square  root  of 

9  a;4  -  6a;3  +  43a;2  -  14a;  +  49. 

8.  Prove  that  (am)n  =  amn. 

a       rr     (ap-q)p +  q  X  (aq)q+r 

bimplifv   * —, — r — - — 

1     J  (ap)p-q 


110  EXAMINATION    PAPERS. 

9.    If  a  and  ft  be  the  roots  of  the  equation  x2—px  +  q  =  0, 
then  will  p  —  a  +  ft  and  q  =  a/3. 
Form  the  equation  whose  roots  are  27  and  — 13. 

10.  When  are  three  quantities  said  to  be  in  continued  pro- 

portion ? 
Show  that  if  x,  (x-\-y),  and  (x  -f-  2y  -f  z)  be  in  con- 
tinued proportion,  then  x,  y,  z  will  also  be  in  con- 
tinued proportion. 

11.  Prove  that  if  x  oc  y  and  y  cc  z,  then  will  x  cc  z. 

Given  that  x  varies  inversely  as  (y*  —  1),  and  is  equal 
to  24  when  y  =  10  ;  find  x  when  y  =  5. 


XXXV. 

University  of  Cambridge,  Eng. 

Previous  Examination,  June,  1881.  —  Time  allowed,  2J  hours. 
Higher  Algebra. 

1.  If  the  first  two  terms  of  an  arithmetical  progression 

are  given,  find  the  sum  of  the  first  n  terms. 
The  sum  of  n  terms  of  an  arithmetical  progression, 
whose  first  two  terms  are  43,  45,  is  equal  to  the  sum 
of  2n  terms  of  another  progression,  whose  first  two 
terms  are  45,  43  ;  find  the  value  of  n. 

2.  Find  the  sum  of  n  terms  of  a  geometrical  progression 

whose  first  term  and  common  ratio  are  given. 
The  sum  of  2w  terms  of  a  geometrical  progression, 
whose  first  term  is  a  and  common  ratio  r,  is  equal  to 
the  sum  of  n  terms  of  a  progression,  whose  first 
term  is  b  and  common  ratio  r2;  prove  that  b  must 
be  equal  to  the  sum  of  the  first  two  terms  of  the 
first  series. 


ALGEBRA.  Ill 


3.  Sum  the  following  series  to  12  terms  : 

(i.)  !-!-¥- 

(ii.)  !  +  *  +  «+ 

(iii.)  1-1.2  +  1.44- 

How  many  strokes  are  struck  in  a  week  by  a  clock 
that  tells  the  hours? 

4.  If  the  sum  of  the  first  n  terms  of  a  series  be  32  n2,  find 

the  rth  term. 

5.  For  what  values  of  m  is  xm  -f-  ym  divisible  by  x  -f-  y  ? 
Divide  a?  +  Sa2b  +  3ab2  +  b3  +  c*  by  a  +  b  +  c. 

6.  A  clock  gains  4  minutes  per  day ;  what  time  should  it 

indicate  at  6  o'clock  in  the  morning  in  order  that  it 
may  be  right  at  7.15  p.m.  on  the  same  day  ? 

7.  The  first  four  nights  of  the  boat-races  both  divisions 

rowed,  and  32  bumps  in  all  were  made.  The  greatest 
number  on  one  evening,  in  the  first  division,  was 
reached  twice,  and  was  equal  to  the  least  number  in 
the  second  division,  which  also  occurred  twice.  This 
number  is  the  middle  one  of  five  consecutive  num- 
bers, of  which  the  first  two  represent  the  number  of 
bumps  the  other  two  nights  in  the  first  division,  and 
the  last  two  represent  the  other  bumps  of  the  second 
division.  How  many  bumps  were  made  in  the  first 
division  ? 

8.  Define  the  logarithm  of  a  number  to  a  given  base.   Prove 

ivyi 

loga-  =  logam  —  \ogan; 
logaMog6a  =  l. 

9.  Find  the  values  of 

log,  a",  log343-v/49,  log3 0.027. 


112  EXAMINATION   PAPERS. 

10.    Having  given  log  2         =0.3010300; 
log  3        =0.4771213; 
log  4.239  =  0.627263; 
Jog  4.24    =0.627366; 
21283i 


find  the  value  of 


103 


XXXVI. 

University  of  Cambridge,  Eng. 

Second  General  Examination  for  the  Ordinary  B.A.  Degree, 
Nov.,  1881.  —  Time  Allowed,  3  hours. 

1.  Solve  the  equations 

M    4a;2+13a;+14  4a; -f- 5 

(       _2^+ll__^      _J 1_ 

v    J  x*-x+l     a?+l     x+S     x-S 
(iii.)  a;-lly  =  l,    Illy -9a;  =  99. 

2.  Find  two  consecutive  numbers,  such  that  the  fourth 

and  eleventh  parts  of  the  less  together  exceed  by  1 
the  fifth  and  ninth  parts  of  the  greater. 

3.  A  certain  number  of  two  digits  is  multiplied  by  4,  and 

the  product  is  less  by  3  than  the  number  formed  by 
inverting  its  digits ;  if  it  be  multiplied  by  5,  the 
tens'  digit  in  the  product  is  greater  by  1,  and  the 
units'  digit  less  by  2  than  the  units'  digit  in  the  orig- 
inal number:  find  the  number. 

4.  Solve  the  equations 

(i.)  3a;2-lla;-4  =  0; 
(ii.)   V7ar+1  =  3+ V2a?-1; 
(iii.)  x2  +  xy  -  2  if  =  -  44,   xy  +  Zy2  =  80. 


ALGEBRA.  113 


5.  If  the  greater  sides  of  a  rectangle  be  diminished  by  3 

yards,  and  the  less  by  1  yard,  its  area  is  halved. 
If  the  greater  be  increased  by  9,  and  the  less  dimin- 
ished by  2,  the  area  is  unaltered ;  find  the  sides. 

6.  If  the  number  of  pence  which  a  dozen  apples  cost  is 

greater  by  2  than  twice  the  number  of  apples  which 
can  be  bought  for  Is.,  how  many  can  be  bought  for 
9  s. 

7.  Define  ratio.     If  a  be  less  than  b,  show  that  a :  b  is  a 

less  ratio  than  a  + 1:5  +  1.  What  is  the  least 
integer  which  must  be  added  to  the  terms  of  the 
ratio  9 :  23,  so  as  to  make  it  greater  than  the  ratio 
7:11? 

8.  The  first  and  fourth  terms  of  a  proportion  are  5  and 

54;  the  sum  of  the  second  and  third  terms  is  51; 
find  them. 

9.  If  A  varies  directly  as  P,  inversely  as  Q,  and  directly 

as  P,  and,  if  when  P=  a,  Q  =  b,  P  =  c,  A  =  abc, 

find  A  when  P=>     Q  =  ™    R  =  a±- 
abc 

10.  Find  the  sum  of  n  terms  of  a  geometrical  progression, 

of  which  the  first  term  is  a  and  the  common  ratio  r. 
Sum: 

64  +  641  + to  29  terms  in  arithmetical  progression; 

64  +  96   + to  7  terms  in  geometrical  progression. 

11.  The  common  difference  of  an  arithmetical  progression 

is  2,  and  the  square  roots  of  the  first,  third,  and 
sixth  terms  are  in  arithmetical  progression;  find  the 
series. 

12.  The  sum  of  four  numbers  in  geometrical  progression 

is  170,  and  the  third  exceeds  the  first  by  30 ;  find 
them.  . 


114  EXAMINATION    PAPERS. 

XXXVII. 

University  of  Cambridge,  Eng. 

Second  Previous  Examination,  Dec,  1881.  —  Time  Allowed,  2\  hours. 

1.  Simplify  6(a-2b)(b-2a)-  (a-Sb)  (46-  a)  -12ab, 

and  from  the  sum  of  (2  a  —  b)2  and  [a  — 2  b)2  take 
the  square  of  2  (a  —  b). 

2.  Define  multiplication,  product,  and  coefficient. 
Divide  14  a4  +  15  a3  b  +  33  a2  b2  +  36  a£3  +  28  b* 

by  7a.2-3a£+1462. 

3.  Find  the  value  of  (a-b)2+(b-c)2+(a-b)(b-c)  +  bc2 

when  a  =  1,  5  =  —  2,  c  =  J. 

4.  Eesolve  into  the  simplest  possible  factors  : 

(i.)  Sof  +  bxy-by2; 

(ii.)  x^-lSx'y  +  ^xy2; 
(iii.)  (a  +  2b  +  3c)2-4(a  +  b-c)2; 
(iv.)  81a;4-625y4. 

5.  Define  the  highest  common  factor  of  two  algebraical 

expressions. 
Find  the  highest  common  factor  of 

7a8- 10a*- 7*+ 10  and  2x3 -x2 -2x-{-l. 

6.  Reduce  to  simple  fractions  in  their  lowest  terms  : 

(1*;    x2  +  bxy  +  6y2    '    x2  +  xy-2y2  ' 

("  \  x~a  |  a2  +  3  q£  .  x -f-  a  t 
x  +  a       a2  —  x2       x  —  a' 
ab  1_1 

,...  x  a  +  6       a2      62 

^•)t-^^xt~t- 

°  +  a2  -  b*      a     b      • 


ALGEBRA.  115 


7.  Solve  the  equations 

{l'}      5  9  7' 

(^1  +  1=14     |  +  |=24; 

(iii.)  bx2-  17  x  +  14  =  0; 

(iv.)  x2+y2  =  ba2+5b2+Sab,   xy=2a2+2b2+5ab. 

8.  Find  the  value  of  xm  X  xn,  when  m  and  w  are  positive 

integers. 
Simplify  a2p+*  X  ap+49r  -f-  a*"*. 

9.  Define  the  antecedent  and  consequent  of  a  ratio. 

If  7  (x  —  y)  =  3  (x  +  y),  what  is  the  ratio  of  x  to  y? 

10.  Show  that  if  a:b::c:d::e:f,  then  will 

a  +  3c  +  2e:a-c::  6  +  3  d  +  2/ :  £  —  /. 

11.  Find  two  numbers  such  that  their  sum,  their  differ- 

ence, and  the  sum  of  their  squares  are  in  the  ratio 
5:3:51. 

12.  Prove  that  if  x  varies  as  -  -f  -,  and  is  equal  to  3  when 

V      * 
y  =  1  and  2  =  2,  then  xyz  =  2  (y  -j- z). 


XXXVIII. 

University  of  Cambridge,  Eng. 

General  Examination  for  the  Ordinary  B.A.  Degree,  June,  1882. 
Time  Allowed,  3  hours. 

1.   Simplify  a      ^      b 

a-\-b     a  —  b 


116  EXAMINATION    PAPERS. 

2.  Solve  the  equations 

w       4  v         ;  7  3     , 

(ii.)  27^-24^-16  =  0; 

(iii.)  2y  +  --4  =  5y  +  -  +  2         --  +  4. 
a;  a?  x 

i 

3.  What  is  the  meaning  of  the  expressions  x~n,  xp  ? 

a.      ,.,     b^Xc"**2*      /iVM+1) 
S-phfy   -p^-  X  y  • 

4.  Solve  the  equations 

(i.)  x2  —  {a  —  b) x  -\-  (a  —  b  -\-  c)  c  =  2cx  -\-  ab  ; 
(ii.)  Vl4rr  +  9  +  2V^  +  l  +  V3#  +  l  =  0; 
(iii.)  3#-2Va^  +  9  =  0,  5V^-3Vy-3  =  0. 

5.  Show  thai^the  sum  of  the  roots  of  the  equation 

ax2  —  bx  +  c  =  0  is  ~ 
If  a  and  ft  are  the  roots  of  the  above  equation,  form  the 

equation  whose  roots  are , 

,  a         P 

6.  When  is  one  quantity  said  to  vary  as  another  ? 

If  ax  +  by  +  1  =  0,  where  a  and  b  are  constant,  and  x 
and  y  are  variable,  and  if  the  values  of  x  are  2  and 
—  9  when  the  values  of  y  are  1  and  —  4,  respectively, 
what  will  be  the  value  of  x  when  y  is  zero  ? 

7.  Define  a  geometrical  progression. 
Find  the  geometrical  mean  of 

9#2  —  12;z  +  4  and  4.r2  -  12 x  -f-  9. 

8.  The  first  term  of  an   arithmetical  progression   is  38, 

and  the  fourth  term  is  86 ;  find  the  sum  of  the  first 
twelve  terms. 
The  first  term  of  a  geometrical  progression  is  27,  and 
the  third  term  is  48 ;   find  the  sum  of  the  first  six 
terms. 


ALGEBRA.  117 


9.    Find,  to  four  places  of  decimals,  the  sum  to  infinity  of 
the  series  1  -\ (-  -i-  + 

vs 

10.  The  perimeter  of  a  rectangular  field  is  306  yards,  and 

the  diagonal  is  117  yards.     What  is  the  area? 

11.  The  expenses  of  a  tram-car  company  are  fixed,  and 

when  it  only  sells  threepenny  tickets  for  the  whole 
journey  it  loses  10  per  cent.  It  then  divides  the 
route  into  two  parts,  selling  twopenny  tickets  for 
each  part,  thereby  gaining  4  per  cent,  and  selling 
3300  tickets  every  week.  How  many  persons  used 
the  cars  weekly  under  the  old  system  ? 

12.  The  price  of  a  passenger's  ticket  on  a  French  railway 

is  proportional  to  the  distance  he  travels ;  he  is 
allowed  25  kilograms  of  luggage  free,  hut  on  every 
kilogram"  beyond  this  amount  he  is  charged  a  sum 
proportional  to  the  distance  he  goes.  If  a  journey 
of  200  miles  with  50  kilograms  of  luggage  cost  25 
francs,  and  a  journey  of  150  miles  with  35  kilograms 
cost  16£  francs,  what  will  a  journey  of  100  miles 
with  100  kilograms  of  luggage  cost? 


University  of  Cambridge,  Eng. 

Previous  Examination,  June,  1882. — Time  Allowed,  2\  hours. — 
Elementary  Algebra. 

1.  Find  the  value  of 

(Za-bb)(a-c)  +  c{2a-c(Za-b)-b2(a-c)\, 
when  a  =  0,  6  =  1,  c  =  —  -J. 

2.  Prove  that 

Multiply  9  a*  xi  +  6  aA  a$  -f  4  a*  by  9  a  —  6  a&  xi  +  4  a* x. 


118  EXAMINATION   PAPERS. 

3.  Resolve  into  their  simplest  possible  factors  : 

\0x2+x-2;  tf2+4;r-4y2+4;  a?-3a2b  +  3ab2-b*+c\ 

4.  What  is  meant  by  a  common  multiple  of  two  quanti- 

ties?    Prove  that  the  sum  of  two  quantities  is  a 
multiple  of  any  of  their  common  measures. 
Find  the  L.C.M.  of 

x3— 3x2y-{-3xy2— 2y3,  xs—x2y—xy2—2y3,  and  x^x^-j-y*. 

5.  Prove  the  rule  for  the  multiplication  of  a  fraction  by 

an  integer. 
Reduce  to  fractions  in  their  lowest  terms  : 
1 


(i) 


2a-3Z> 


2a-Zb-  .  2a-db 

{    J   \x  +  y      x't  +  tf  J  ^  Xy^x      1J 

6.  Prove  that  a  quadratic  equation  cannot  have  more  than 

two  roots. 
Solve  the  equations 

(ii.)  a (2x-y)+b(2x+y)  =  c(2x-y)+d(2x+y)  =  1 ; 
(iii.)  x2-  11^-42  =  0; 

(iv.)  1 |_6  =  -JL__i-_ 2_. 

V     ;  0-1)0-2)  s-2      ar-1 

7.  A   certain  number,  consisting  of  two  digits,  becomes 

110  when  the  number  obtained  by  reversing  the 
digits  is  added  to  it ;  also  the  first  number  exceeds 
unity  by  five  times  the  excess  of  the  second  number 
over  unity.     What  is  the  number  ? 

8.  Define  a  third  proportional  to  two  quantities.     Having 

given  a  third  proportional  to  a  and  b,  and  also  to  b 
and  a,  determine  a  and  b  in  terms  of  them. 


ALGEBRA.  119 

If  a:b  : :  c:  d,  and  x  be  a  third  proportional  to  a  and  c, 
and  y  to  b  and  c?,  prove  that  the  third  proportional 
to  x  and  y  is  equal  to  that  to  a  and  e?. 

9.    "When  is  a  quantity  said  to  vary  inversely  as  another  ? 
If  a  and  b  each  vary  inversely  as  c,  prove  that  the 
sum  of  any  given  multiples  of  a  and  b  varies  in- 
versely as  any  given  multiple  of  e. 

Given  that  x  —  y  varies  inversely  as  z-f -,  and  x-\-y 

1  .  z 

inversely  as  z ;  find  the  relation  between  x  and  z, 

provided  that  x  —  1  and  y  =  3,  when  z  =  ^. 


XL. 

University  of  Cambridge,  Eng. 

Previous  Examination,  June,  1882.  —  Time  Allowed,  2}  hours. 
Higher  Algebra. 

1.  A  has  twice  as  many  pennies  as  shillings ;  B,  who  has 

8d.  more  than  A,  has  twice  as  many  shillings  as 
pennies ;  together  they  have  one  more  penny  than 
they  have  shillings.     How  much  has  each  ? 

2.  A  man  can  walk  a  certain  distance  in  4  hours ;  if  he 

were  to  increase  his  rate  by  one-fifteenth,  he  could 
walk  one  mile  more  in  that  time.    What  is  his  rate? 

3.  Solve  the  equations 

(i.)   V5^  +  l  =  2+V^Tl.       ' 
(ii.)   x2-2y2  =  7,    2x  +  y  =  7. 

4.  A  man  buys  a  number  of  articles,  for  £1,  and  makes 

£1  Is.  Qd.  by  selling  all  but  two  at  2d.  apiece 
more  than  they  cost,     How  many  did  he  buy  ? 


120  EXAMINATION   PAPEES. 

5.  Find  the  sum  of  n  terms  of  the  progression 

a  +  ar  -{-  ar2  -{- 

Find  the  sum  of  10  terms  of  the  progression 
64  +  96  +  144  + 

6.  The  fifth  term  of  an  arithmetical  progression  is  81,  and 

the  second  term  is  24 ;  find  the  series. 

7.  Find  an  arithmetical  progression  whose  first  term  is  3, 

such  that  its  second,  fourth,  and  eighth  terms  may 
be  in  geometrical  progression. 

8.  If  a  :  x  : :  b  :  y  : :  c  :  z,  prove  that 

la3  +  mb2  y  +  ncz2     pa  +  qb  +  re 
la1  x  +  mby2  +  nz3     px  +  qy  +  rz 

9.  Prove  that  loga  (pq)  =  logap  +  loga  q. 

10.    Having  given  log102  =  0.3010300,  find  the  logarithm 
to  base  10  of  25,  0.03125,  and  (0.025)i 


XLI. 

Univeesity  of  Oxfoed,  Eng. 

Local  Examination,  Junior  Candidates,  May,  1880.  —  Time  Allowed, 
2}  hours. 

No  credit  will  be  given  for  any  answer,  the  full  working  of  which 
is  not  shown. 

I.   Algebea. 

t     v   jx-l        i        ,a2-c-3ac(b-2c)         L        T+b 

1.  Find  the  value  of j— ; — prt L  +  \2a 7"' 

when  a  =  1,  b  —  0,  and  c  —  —  \. 

2.  Multiply  x*  —  axz  +  a?x  —  a*  by  x2  +  ax  +  a2 ;  also  di- 

vide p*  -  9pq"  +  18^4  by  p2  -  Zpq  +  3  q2. 


ALGEBRA.  121 


3.  Simplify 

(i)  (2x+y _    v   V  x X+A- 

\     y  2x  +  y)\x  +  y         x    J' 

(^  _i oS  +  2    ,     s  +  3 

V   ';  x  +  1         x2-l^ {x-yY 

4.  Find  the  G-.C.M.  of^3-6^-4and  Sx3  —  8x  +  8 

the  L.C.M.  of  (3  a2  -  S"ab)2,  18  (a?  b2  -  ab4),  and 
24(W-66). 

5.  Solve  the  equations 

(i.)  5^3_,(;,_24)  =  2^1  +  1| 

(iL)S=22lSV6(*+«=n(y+5)- 

II.   Higher  Algebra. 

6.  Solve  the  equations 

rn     2*       10^+1_3. 

(ii.)  a(a-6)a;2  +  6(>  +  6);r-262=:0; 
(iii.)V  +  2ay=|,   0^-4^=1^ 

7.  The  sum  of  2  numbers  is  35 ;  and  their  difference  ex- 

ceeds one-fifth  of  the  smaller  number  by  2 ;  find  the 
numbers. 

8.  After  £12  have  been  divided  equally  among  a  certain 

number  of  men,  an  additional  shilling  apiece  is  given 
to  them  ;  and  it  is  then  found  that  each  possesses  as 
many  shillings  as  there  are  men.  Find  the  number 
of  the  men. 

9.  Prove  that  if  b  be  a  mean  proportional  between  a  and 

c,  thena2  +  2b2:a::b2  +  2c2:c. 
10.    Sum  to  6  terms  the  series  J  +  1\  +  3-J-  -f Also  in- 
sert 12  arithmetic  means  between  —\  and  5. 


122  EXAMINATION    PAPERS. 

XLII. 

University  of  Oxford,  Eng. 

Local  Examination,  Senior  Candidates,  May,  1880.  —  Time  Allowed, 
2 £  hours.  * 

No  credit  will  be  given  for  any  answer,  the  full  working  of  which 
is  not  shown.  . 

Candidates  are  reminded  that  in  order  to  pass  in  mathematics  they 
must  satisfy  the  Examiners  in  the  first  part  of  this  paper. 

I.     Algebra  to  Quadratic  Equations. 

1.  If  m  = ■  an(i  n  = ^ >  find  the  value  of 

2  2 

m2  -f  n\ 

2.  Find  the  G.C.M.  of  2^+3^-7^-10,  5-9^-4^+4^; 

and  the  L.C.M.  of  a2  +  ab,  b2  -f-  ab,  ab. 

3.  Simplify -J- -?+l 

x—1       X       X  +  1 

4.  Find  the  square  root  of  4a;4  -  20#3  +  13^  +  30#  -f  9. 

5.  Multiply  xi  +  2  a*  +  2  by  xi  -  2x1  +  2,  and  divide 

a2b~2  +  b2a~2  -f  1  by  ai"1  +  6a"1  -  1. 

6.  Solve  the  equations 

i;i-s     20' 

(ii.)  *  =  (2-ar)(2  +  a0; 
/•••  N  a      x      b      x 

(ill.)   -  _!_-  =-  4--  • 

•    v     ;  x^a      x^b' 
(iv.)  x2  +  y2  =  169  =  hx  +  12y. 

7.  Two  rectangular  fields  each  contain  one  acre ;  one  of 

the  fields  is  four  poles  shorter  and  two  poles  broader 
than  the  other.  Find  the  length  and  breadth  of 
each  field. 


ALGEBRA.  123 


II.    Higher  Algebra. 

8.  A  spends  £a  in  buying  a  number  of  articles,  all  at  the 

same  price ;  B  spends  £  b  in  the  same  way,  except 
that  he  buys  n  more  articles  than  A  buys,  and  pays 
£c  less  for  each.  Find  an  equation  to  determine 
the  number  of  articles  bought  by  A. 

9.  Solve  the  equations 

(i.)  4*+l  =  5x2*-1; 
(ii.)  x*  -  Sx*  -  2x2  -  Sx  +  1  =  0  ; 
(iii.)  Sx  +  4y  =  23  (in  positive  integers). 

10.  Prove  the  formulae  for  finding  the  nth.  term  and  the 

sum  of  n  terms  of  an  arithmetic  progression,  the 
first  term  and  the  common  difference  being  known. 
A  man  pays  his  gardener  15  s.  a  week  for  the  first  fort- 
night ;  at  the  end  of  the  first  and  of  every  succeed- 
ing fortnight  he  raises  the  wages  6  d.  per  week. 
What  will  the  gardener  have  received  in  all  at  the 
end  of  fifty  weeks  ? 

11.  Find  the  cost  of  an  annuity  of  £A  per  annum,  to  be 

paid  quarterly,  and  to  continue  for  p  years ;  the  first 
payment  to  be  made  at  the  end  of  the  first  quarter, 
reckoning  compound  interest,  at  the  rate  of  £r  per 
cent  per  annum,  to  be  due  at  the  date  of  each  quar- 
terly payment. 

12.  Enunciate  the  binomial  theorem. 

Show  that  the  coefficient  of  the  middle  term  in  the  ex- 
pansion of  (1  +  x)2n  is  the  sum  of  the  coefficients  of 
the  two  middle  terms  in  the  expansion  of  (l+^)2w_1. 

13.  Prove  that 


1    ,   1 


00    1+,T+r^+ =1+,t+^+ 


x   .  x 


Li  |2      ;       ii  [2 


/••  n  i  ll  -\-  x      x  ,  xz  .  xh  , 


124  EXAMINATION    PAPERS. 

XLIII. 

University  of  Oxford,  Eng. 

Local  Examination,  Junior  Candidates,  June,  1881.  —  Time  Allowed, 
1\  hours. 

No  credit  will  be  given  for  any  answer,  the  full  working  of  which 
is  not  shown. 

I.    Algebra. 

1.  Find  the  value  of  a3  +  b3  -  c3  +  3  abc : 

(L)  when  a  =  J,  £  —  3,  c  =  $; 
(ii.)  when  c  =  a  +  b. 

2.  Multiply  together   x2  —  7a; +,6,    x2  +  7x  —  18,    x3  —  1, 

and  express  the  result  in  simple  factors. 

3.  Find  the  G.C.M.  of  2x*+  1  x2  +  10a;  +  5  and  x3  +  Sx> 

+  4^  +  2,andtheL.O.M.of6^y2(^  +  2/),3^3(^-3/)2, 
and  4  (x2  —  y2). 

4.  Simplify 


(^2  +  2/2)      ^(^2  +  3/2)      ayy 


5.  Solve  the  equations 

,.  v  I  —  rg  .  2_ff  .  #— 1 

W      2     +     f         4+     3     ' 
(ii.)  9^-8y  =  l,  12s  — 10y  =  l. 

6.  Into  a  cistern  one-third  full  of  water  31  gallons  are 

poured,  and  the  cistern  is  then  found  to  be  half  full ; 
find  its  capacity. 


ALGEBRA.  125 


II.     Higher  Algebra. 

7.  Solve  the  equations 

(i.)  x  =  ^+^; 
K  J         64^49' 

(ii.)  ax2 =  cx  —  bx2; 

v    J  a  +  b 

(iii.)^+*±2  =  o,*j,  =  i. 

8.  A  person  bought  a  certain  number  of  sheep  for  £210. 

He  lost  10,  and  to  make  up  the  deficiency  sold  the 
remainder  at  10 s.  profit  per  head.  How  many  did 
he  buy  ? 

9.  Prove  the  rule  for  the  summation  to  infinity  of  a  geo- 

metrical progression ;  and  sum  to  n  terms  and  to 
infinity  Si  +  5  +  3  + 

10.  The  seventh  term  of  an  arithmetical  progression  is  1 ; 

and  the  sum  of  twenty-five  terms  is  zero.  Find  the 
progression. 

11.  If  a  :  b  : :  c  :  d,  prove  that  a-\-b:a  —  b  \\  c-\- d:c  —  d. 

12.  If  2x  +  Zy  :2x-Zy  ::2a2  +  U2 :2a2  -Zb2,   then    x 

has  to  y  \ho,  duplicate  ratio  that  a  has  to  b. 

XLIV. 

University  of  Oxford,  Eng. 

Local  Examination,  Senior  Candidates,  June,  1881.  —  Time  Allowed, 
2\  hours. 

No  credit  will  be  given  for  any  answer,  the  full  working  of  which 
is  not  shown. 

I.    Algebra  to  Quadratic  Equations. 
1.    Prove  that 

(a+b)(a+x)(b  +  x)-a(bi-xy-b(a+xy=(a-b)2x, 
and  divide  a3  —  b3  by  a*  —  2ab*  +  2ahb  —  M. 


126  EXAMINATION    PAPERS. 

2.  Resolve  into  component  factors 

(i.)63aty-28ay;     (ii.)  a5-  a4b  -  ab*  +  b*. 
Find  the  remainder  when  an  +  bn  is  divided  by  a  —  b. 

3.  Find  the  G.C.M.  of 

x4-6^3+13^2-12.r+4  and  ^4- 4 ^+8^-16^+ 16, 
and  the  L.C.M.   of   x?  —  if,    ar'-J-y3,  x3  —  xy2,    and 

4.  Simplify  the  fractions 
px a;  +  2     ^+1         2 

(ii.) 
Cm.) 


ar+1 

1 

x  + 

1 

2 

x  -f4' 

1 

I-1 

1 
x* 

1 
a? 

1         -^ 

a; 

*-fcl 

Va;  +  Va;  — 

1 

V#  —  Va-  —  i 

Var  —  Va?  —  1      Va-  +  Va;—  1 

5.  Solve  the  equations 

(i.)  iK2^-32)-(^-hl6)}=1irS(^_20)-(2^-ll)S; 
(ii.)  (a;+5)(y+7)  =  (ar+l)(y-9)-j-112, 2*+5  =  3y-4; 

V     }         a^  +  5 

6.  A  person  invests  £500,  part  of  it  at  5  per  cent  and 

the  remainder  at  3  per  cent ;  and  he  thus  gets  4? 
per  cent  on  the  whole.  How  much  does  he  invest 
at  each  rate  of  interest? 

7.  Find  the  square  root  of  9-24ar-68a72+112a;3+196^4. 

II.     Higher  Algebea. 

8.  Prove  that  a  ratio  of  greater  inequality  is  diminished, 

and  a  ratio  of  less  inequality  is  increased,  by  add- 
ing the  same  number  to  each  of  its  terms. 
If  a:  a  —  b  ::  c  :  c  —  d,  then  a-\-b  :  b  ::<?  +  d:d. 


ALGEBRA.  127 

9.    Prove  that  the  geometrical  mean  between  two  numbers 
is  also  the  geometrical  mean  between  the  arithmeti- 
cal and  harmonical  means. 
Sum  the  series  : 

(i.)  25i+24+22|+21  + to  15  and  to  21  terms  ; 

(ii.)  4j*g-+2f +1-J-J •  to  n  terms,  and  to  infinity. 

10.  Prove  that  the  number  of  combinations  of  n  things 

taken  r  together  is  equal  to  that  of  n  things  taken 
n  —  r  together,  and  greater  than  that  of  n  —  1  things 
taken  r  —  1  together. 
How  many  different  numbers  can  be  made  with  all  or 
any  of  the  figures  of  the  number  1881  ? 

11.  Employ  the  binomial  theorem  to  expand  (a  —  x)n  ;  also 

(a  +  x)%  in  ascending  powers  of  x  to  5  terms. 

12.  If  a,  ft  are  the  roots  of  the  equation  ax*  +  bx-\-  c  =  0, 

find  the  values  of  a+/J  and  a3+/?3  in  terms  of  a,  b,  c. 

13.  Solve  the  equations 

(i.)*3-i  +  7(*3  +  l)  =  0; 
(ii.)  3^2  +  y2  =  3^+7  =  19. 

XLV. 

University  of  Oxford,  Eng. 

Local  Examination,  Junior  Candidates,  June,  1882.  —  Time  Allowed, 
2$  hours. 

No   credit  will  be   given  for  any  answer,  the  full  working  of 
which  is  not  shown. 

I.   Algebea. 


1.    Find  the  value  of 


x2  —  yl 
x2  +  yl 


/•  \  __r  a-\-b  a  — 

(i.)  when  x  = -J-,  y  =  ~2 

(ii.)  when  a?  =  J,  y  =  -  \. 


128 


EXAMINATION   PAPERS. 


2.  Square  2b  —  3c,  and  find  the  product  of  a-j-2b  —  3c 

and  a  —  2b  +  3  c. 

3.  Find  the  G.C.M.  of  2  a56  +  2  a2  6*  and  4a5  +  4  a362  +  4  a£4 ; 

and  the  L.C.M.  of  5x(x2  +  2x+l),  10s*  (a*— 1), 
and  15(>+  1)  (*2-  2^  +  1)- 

4.  Simplify 


(i.) 
(ii.) 

1 
2 

2x 

5- 

x  _ 

.5-*      1 

x  — 

^2 

2 
=  4, 

ar-2      2 

2y  +  |  = 

1. 


A  and  B  set  out  at  the  same  time  from  the  same  spot 
to  walk  to  a  place  6  miles  distant  and  back  again. 
After  walking  for  2  hours,  A  meets  B  coming  back. 
Supposing  B  to  walk  twice  as  fast  as  A,  and  each 
to  maintain  uniform  speed  throughout,  find  their 
respective  rates  of  walking. 


II.  Higher  Algebea. 
Solve  the  equations 


« 1[ 

(iii.) 


(x  +  2)(2x- 

x  +  y .,; 3  ^ 


1)     0-2)0 
-9y2=16 


(2o;+l)J: 


Ax2 


x  —  y 

8.   Show  that  the  sum  of  any  two  consecutive  whole  num- 
bers is  equal  to  the  difference  of  their  squares. 


ALGEBRA.  129 


9.  Find  the  sum  to  n  terms  of  an  arithmetical  progres- 
sion, the  first  term  and  the  common  difference  being 
given. 
What  is  the  amount  of  a  debt  which  can  be  discharged 
in  two  years  by  the  payment  of  10  s.  the  first  month, 
£1  the  second,  30s.  the  third,  and  so  on,  no  interest 
being  exacted  ? 

10.  Sum  to  n  terms,  and  to  infinity,  the  series  |- — -J-  -f  f  — 

11.  If  Oy :  bx : :  a* :  b2,  prove  that 

(a,  +  a2)2 :  (bx  +  b2f : :  a,2+  a* :  W  +  W. 

12.  Show  that  if  b  is  a  mean  proportional  between  a  and  c, 

then  (a2  +  V)  (b2  -f  c2)  =  (ab  +  be)2. 

XL  VI. 

University  of  Oxford,  Eng. 

Local  Examination,  Senior  Candidates,  June,  1882.  —  Time  Allowed, 
2\  hours.  ' 

No  credit  will  be  given  for  any  answer,  the  full  working  of 
which  is  not  shown. 

Candidates  are  reminded  that  in  order  to  pass  in  mathematics, 
they  must  satisfy  the  examiners  in  the  first  part  of  this  paper. 

I.   Algebra  to  Quadratic  Equations. 

1.  Prove  that  (a  +  2  bf  =  a?  +  2  bz  +  6  b  .  (a  +  b)\  and  find 

the  quotient  a2  +  aU  +  b  -f-  a  +  ah  bi  -f  b?. 

2.  Simplify 


I+-7-  1+  1      ' 


y  +  z  z  +  x  x  +  y 

&-*)'      ,r  (c-a)V      . 


(iii)       ^-^       J  '     ('-")'      -.-      (a~*) 


130  EXAMINATION   PAPERS. 

3.  Find  the   G.C.M.   of  3^  +  17a;2  +  22*  +  8  and  6x3 

+  25#2  +  23*  +  6,  and  the  L'.C.M.  of  (V  -  yj, 
xz  +  y3,  xs  —  y3,  and  (x2  —  y2)2. 

4.  Extract  the  square  root  of 

(i.)  7-4 V3; 
(ii.)  x*(xz  +  2)  +  2*2(*2  +  1)  -  *(*  -  2)  +  1. 

5.  Solve  the  equations 

(i.)  ll*-5(*-6)-6(3*-ll)  +  9(*-7)  =  0; 

^    ;  8^6      12^4 
(iii.)  (3  x  -  6)2  +  7  x*  -  256  =  0. 

6.  My  income  of  £240  is  derived  from  two  sums  invested 

at  three  and  nine  per  cent  respectively ;  but,  if  the 
rates  of  interest  were  interchanged,  my  income  would 
be  doubled  ;  find  the  sums  invested. 

II.   Higher  Algebea. 

7.  If  a  :  b  : :  c  :  d,  prove  that 

(i.)  a3  +  d3>b3  +  cz] 
(ii.)  Sa+2b:Sa  —  2b::Sc  +  2d:Sc  —  2d. 

8.  Find  an  expression  for  the  sum  of  n  terms  of  a  geo- 

metrical series,  and  explain  the  expression  "  sum  to 
infinity." 
Find  the  fourth  term  in 

(i.)  2  +  21  +  3    + 

(ii.)  2  +  21  +  31+ 

(iii.)  2  +  21  +  31+ 

9.  Find  the  number  of  permutations  nPr  of  n  things  taken 

r  together.     If  "P4 :  —*jp«  : :  3  :  22,  find  n. 

10.    Show  that  the  number  of  terms  in  the  expansion  of 
(a  +  x)n  is  n  +  1,  if  n  is  a  positive  integer. 
Apply  the  binomial  theorem  to  find  (10.001)7  to  five 
places  of  decimals. 


ALGEBRA.  131 


11.  If  a,  /3  are  the  roots  of  ax2  +  bx  +  c  =  0,  find  the  equa- 

tion whose  roots  are 

(i.)    a\  F;         (ii.)«(l+/B),/8(l+a). 

12.  Solve  the  equations 

(i.)  144  a*  -  1  +  6  -Vdx2  -  x  =  16  a? ; 
(ii.)  x2  (s»-y»)  =  25,   y»(s»  +  y2)  =  19| ; 

(iii.)    </U-X+VU  +  X  =  4:. 

XL  VII. 

University  of  Oxford,  Eng. 

.Krsi  Examination  of  Women,  May,  1880.  —  2$me  Allowed,  2\  hours. 

1.  Find  the  value  of  — - — ^— —  when  a  =  l,b  —  —  %,c  =  Q. 

a3  +  463 

2.  Take  a  +  2b  +  Sc  —  4d—5e  from  3ct-46 +  c  —  c7  +  e. 

3.  Multiply  a3  -  a2  6  +  ah2  -b3  by  a  -f-  b, 

and  divide  a4  -f  a2  52  +  &4  by  a2  +  a£  +  b2. 

4.  Find  the  H.O.F.  of  a4  +  5  a2- 6  and  a4  +  5  a3 +  4,  and 

the  L.C.M.  of  12(a3-63),  15(a3+6>3),  20ab(a2-b2). 

5.  Simplify  "a  +  4*  +  3 

1     J     a2 -a- 2 

6.  Extract  the  square  root  of  a4  —  2a3  +  2a2  —  a  +  J. 

7.  Solve 

(i.)  70-1) -60-2)  =  3(^-3); 

(ii.)  1  +  1+1=*- 7; 

(iii.)  (x  -  3)  0  -  13)  ==  (a  -  4)  0  -  9) ; 
(iv.)  8rs  +  3y=74,   9^-2y  =  51. 

8.  I  have  in  my  purse  £1  13  s.  9  c?.  made  up  of  a  certain 

number  of  pence,  twice  the  same  number  of  farth- 
ings, and  thrice  the  same  number  of  fourpenny 
pieces.     Find  the  number  of  each  coin. 


132  EXAMINATION    PAPERS. 

9.    A  is  thrice  as  old  as  B.     Seven  years  ago  A  was  four 
times  as  old  as  B.     Find  their  ages  now. 

10.  A  and  B  play  at  cards.  A  wins  six  shillings,  and  finds 
he  has  thrice  as  much  as  B.  The  game  is  continued 
till  A  finds  he  has  lost  twenty-four  shillings,  and 
then  has  a  third  of  what  B  has.  With  what  sum 
did  each  begin  ? 


XLVIII. 

University  of  Oxford,  Eng-. 

Second  Examination  of  Women,  May,  1880.  —  Time  Allowed,  2  hours. 

1.  Divide  a*  —  a~*  by  oh  —  a~%. 

2.  Simplify 

(6a2-a-2)(8a2--10a  +  3)(12a2  +  17a  +  6), 
(8a2  -  2a  -  3)  (12a2  +  a  -  6)  (6a2  +  a  -  2)     ' 
6a2-17a+12    .  27aM-18a-24  ,  25a2-25a+6 
12a2-25a+12i_12a2+7a-12  +  20a2-23a+6* 

3.  Extract  the  square  root  of 

(i.)  4a2+-2-ll-6a-1  +  4a; 
a 

(ii.)  41-12V5. 

4.  Solve  the  equations 

v         x  +  b         x-\-a 
(ii.)  ^-7*4-10  =  0; 
(iii.)  V*  -  2  -f-  s/x  +  3  =  V±x  -j-  1  ; 
(iv.)  x  +  y  =  8,   ay  =15. 

5.  If  a  :  b  : :  e :  d,  prove  that  a  -\-b  :  a  —  b  : :  c  +  d :  c  —  d. 


ALGEBRA.  ISi 


6.  £21   18  6-.  is  divided  equally  among  a  certain  number 

of  persons.  If  each  received  a  penny  less,  each 
would  have  had  as  many  pence  as  there  were  per- 
sons.    Find  the  number  of  persons. 

7.  A  steamer  takes  2  hours  24  minutes  less  time  to  travel 

from  A  to  B  than  from  B  to  A.  The  steamer  trav- 
els at  the  rate  of  16  miles  an  hour,  and  the  stream 
flows  at  the  rate  of  6  miles  an  hour.  Find  the  dis- 
tance of  A  from  B. 


XLIX. 

University  of  Oxford,  Eng. 

First  Examination  of  Women,  June,  1882. —  Time  Allowed,  2 \  hours. 

1.  Find  the  value  of  x{y-\-z)-\-y[x—  (y-fz)]  —  z[y— x{z— x)] 

when  x  =  3,  y  =  %  2  =  1. 

2.  Subtract  2  a*  +  fa*  —  x  +  \  from  |  x*  +  x2  -  \ x  -  f . 

3.  Multiply  l  +  2x+3y+4:X2-6xy  +  9y2  by  l-2x-Sy. 

4.  Divide  x* --^x3  +  x2 +  %x -2  by  #-■§. 

5.  Kesolve  into  factors 

(i.)  4^-36^22/2; 
(ii.)  (2x-3y)2-(x-2y)\ 

6.  Find  the  G.C.M.  of  x3  -  3x  +  2  and  x3  +  4#2  -  5. 

7.  Find  the  L.C.M.  of 

12(1  -x2),  15(1 -a;)2,  and  20(a?  +  s»). 

8.  Simplify 
a2h  _  c2d  _  4:ab  _  bed  —  cd2 


W-x^x^x^ 


cd      U2~Ud 


W  Vvl  +  4^"rl-4^  '  Vl-4a?      1  +  4^ 


134  EXAMINATION    PAPEES. 

9.    Solve  the  equations 

W  3^  9       27        11 

fin  x+a a- x+h-2- 

2*-_3  2y+_3  = 

y  x 

10.  A  person  walks  a  certain  distance  at  the  rate  of  3? 

miles  an  hour,  and  finds  that  if  he  had  walked 
4  miles  an  hour,  he  would  have  gone  the  same  dis- 
tance in  less  time  by  one  hour;  what  is  the  distance? 

11.  Find  two  numbers  such  that,  if  half  the  first  be  added 

to  the  second,  or  \  of  the  second  be  added  to  the 
first,  the  sum  will  in  either  case  be  30. 

12.  Find  the  square  root  of 

a6  -  4a5  +  8a4-  10a3  +  8a2  -4a  +  1. 


L. 

Univeesity  of  Oxfoed,  Eng. 

Second  Examination  of  Women,  June,  1882.  —  Time  Allowed,  2  hours. 

1.  Find  the  value  of  (a  +  &-c)2+(Z>+c-a)2  +  0+a-6)2 

when  a  =  2,  b  =  3,  c  =  —  £. 

2.  If  a  +  b  +  c  =p,  be  +  ca  +  oh  =  q2,  and  abc  =  r3,  prove 

that  a3  +  bz  +  c3  =pz  -  Spy*  +  3  r3. 

3.  Find  the  G.C.M.  of      - 

2  a*  +  x2  _  12x  +  9  an(j  2a:3  -7x2  +  l2x-  9,  and 

1 ; 1 

2s8  -  7a?  +  12a? -9     $af+y— 1&*  +  9 

.      A  , ,  .       ii         rr  — a         x4-a  ,      2a;  (3  a  — a;) 

4.  Add  together  -; — - — ^,    -f c-2,   and 


simplify 


(x  -f-  a)2'    (a;  —  a)2'  (a;  —  a)  (x  +  a)2 


ALGEBRA.  135 


5.  Find  the  square  root  of 

x6  +  aV  +  2a? x*  +  la'x2  -  d'x  +  a\ 

6.  Solve  the  equations 

(,)    '    •    2       1 


2a; -1      4:x-'d      x—  1' 


(ii.)V53*  +  a?-16  =  3a?-2; 
(iii.)  (a:  —  y)  (a?  —  3y)  =  24,  a;  — 2y  =  5. 

7.  There  are  250  flowers  in  a  conservatory ;  the  number 

of  geraniums  is  five  times  the  number  of  roses,  and 
is  less  by  30  than  the  number  of  other  flowers.  How 
many  roses,  geraniums,  and  other  flowers  respec- 
tively are  there  in  the  conservatory  ? 

8.  Find  the  value  of  x  in  each  of  the  following  propor- 

tions : 

(i.)  1 :  x  : :  2  :  3  ;    (ii.)  1  :  2  : :  3  :  x ;    (iii.)  4  :  x  : :  x  :  9. 

9.  Two  persons,  A  and  B,  start  at  the  same  time  by  their 

watches,  from  two  places  24  miles  apart,  and  drive 
towards  each  other  at  rates  which  are  as  2  :  3 ;  but 
in  consequence  of  A's  watch  being  \  hour  too  fast, 
and  B's  i  hour  too  slow,  they  meet  half  way.  At 
what  rate  does  each  drive  ? 


LI. 

University  of  Oxford,  Eng. 
First  Examination  of  Women,  June,  1883.  —  Time  Allowed,  2\  hours. 

1.  Evaluate   (x  -  yf  +  (y  -  zf  +  (z  -  x)\  when   x  =  3%, 

y  =  2*jZ=tt. 

2.  From  the  sum  of  ^(2a?-3y-j-4z)  and  ^  (4a;  +  3?/ -llz) 

subtract  -fa(8x  —  9y  -f-  62). 


136  EXAMINATION    PAPERS.  • 

3.  Multiply  x2  +  y2  +  a  (x  —  y)  by  xy  —  a  (x  +  y)  +  a*. 

4.  Divide  a*  +  Sy3  -  125 z3  +  30#yz  by  a;  -f  2y  —  5 z. 

5.  Express  in  factors 

(i.)  7  a* -77a; -182; 
(ii.)  20  a;4- 60  ar5?/ +  45  a;2  y2. 

6.  Find  the  G.C.M.  of  a*+lla;  +  30,  9a;3+  53a;2-9a;-18. 

7.  Find  the  L.G.M.  of 

15a;2(a2-2aa;+a;2),  21a2(a2+2aa;+a;2),  35  aa<a2- a;2). 

8.  Simplify   a~b      a  +  b  and  find  the  value  of 

1 


a—b  a-\-b 
a 

•    a—b  a—b 

x  +  2  :x-2  4 


when  x=  i. 


2-x      2  +  x      4:-X2 
9.    Solve  the  equations 

(i.)  f(*-l)  +  3(|-9)-(*-13)  =  ll; 

(ii.)  JL  +    1 ^  =  0; 

x—a      x—b      x—a—b 

,...  N  5a:  +  7y      7a;-J-5y      Q 
(m.)  — 1^  =  — ^-^  =  8- 

10.  The  sum  of  three  consecutive  whole  numbers  exceeds 

the  greatest  of  them  by  19 ;  what  are  the  numbers? 

11.  Find  the  fraction  which  is  such  that  if  3  be  subtracted 

from  the  numerator,  and  ,5  added  to  the  denomina- 
tor, the  value  is  \ ;  but  if  5  be  subtracted  from  the 
numerator,  and  3  added  to  the  denominator,  the 
value  is  -J. 

12.  Extract  the  square  root  of 

36a;2  -I20ax-  12a2  x  +  100a2  -f-  20a3  +  a\ 


ALGEBRA.  137 


LII. 

University  of  Oxford,  Eng. 

Second  Examination  of  Women,  June,  1883.  —  Time  Allowed,  2  hours. 

1.  Divide  x*  —  if  —  xy (x*  -\-2f)  by  (x  -\-  y)  (x  —  y)  —  xy, 

and  verify  the  result  when  x  =  2y  =  |. 

2.  Prove  that  (cy  —  hzf  -f  (az  —  ex)2  -f  (bx  —  ay)2  =  x2-\-y2 

+  z2  -  (ax  +  by  +  cz)2  if  a2  +  b2  +  c2  =  1 . 

3.  Find  the  G.C.M.  and  the  L.C.M.  of 

7x3  -2x2-b  and  7*3  +  I2x2  +  10  #  +  5. 

4.  Simplify 

*0+y)     O^'V  +  zy+y*)  '  O+y)2' 
(ii.) t . * ■    2ab  . 

5.  Find  the  square  root  of 

(^  +  y2)(^  +  22)  +  2^(^  +  7/2)(3/  +  2)+4^yZ. 

6.  Solve  the  equations 

r  ,  x  —  3      2x  —  l  .  7x  —  11      fi 

(l-}-2f 3T  +  -5|-  =  6' 

'..\  5(Sx2-l)      2 

(iii.)  ^2  +  y2  =  9,    a;-3y  +  3  =  0. 

7.  A  number  of  men  are  employed  at  13  s.  6d.  per  week 

each,  and  women  at  10s.  6d. ;  the  number  of  women 
exceeds  that  of  the  men  by  6,  but  the  men's  wages 
amount  to  27s.  per  week  more  than  the  women's; 
how  many  of  each  sex  are  employed  ? 


138  EXAMINATION    PAPERS. 

8.  Find  the  mean  proportional  between  *£  and  f ;  and 

the  fourth  proportional  to  9,  11,  27. 
If  the  first  two  terms  of  a  proportion  are  3  and  2,  and 
the  third  term  exceeds  the  fourth  by  5,  what  are 
the  third  and  fourth  terms  ? 

9.  Divide  £500  among  three  persons,  so  that  the  share  of 

the  first  may  be  to  that  of  the  second  as  8  :  17,  and 
the  share  of  the  third  to  the  other  two  together  as 
3:2. 

LIII. 

University  of  London,  Eng. 

Matriculation  Examination,  Arithmetic  and  Algebra,  June,  1877. 

1.  Divide  ff +  6f-72V  hJ  5f-3T  +  TT>  and  multiply 

the  result  by  6^  +  $£>  -  1% 

2.  Find  what  fraction  of  a  guinea  is  equal  to  the  differ- 

ence between  f  of  a  crown  and  |f  of  a  shilling. 

3.  Calculate  to  five  places  of  decimals  the  fraction 

3.70271  x  0.64732 
0.043679 

4.  Reduce  the  circulating  decimal  1.52372  to  a  vulgar 

fraction  in  its  lowest  terms. 

5.  Extract  the  square  roots,  to  five  places  of  decimals,  of  the 

numbers,  3.9726523,  0.39726523,  and  0.039726523. 

6.  Simplify  the  algebraic  expression, 

_b_ 1 

Sax— 5  by      ax  2  by'1 

T~V~Zax-2>byk 

7.  Divide  15a;5-  17a?*- 24a3  +  138a;2  -  130a;  +  63  by 

5  x3  +  6  x2  —  9  x  -f-  7,  and  verify  the  result. 

o    -^  xi    .  -oX2  —  xy-\-y2      a;3  +  ?/      v    ,,       x      u 

8.  Prove  that  if  -= y      y.  =    .  \  %  X  -  then  -  =  - 

u2  —  uv-\-v2     uijrv6     y  y     v 


ALGEBRA.  139 


9.  Given  the  first  term,  the  middle  term,  and  the  number 
of  terms  in  an  arithmetical  progression ;  find  the 
sum  of  the  series.  Has  this  problem  a  meaning,  if 
the  number  of  terms  is  even  ? 

10.  Find  to  n  terms  and  to  infinity  the  sum  of  the  geomet- 

rical progression  in  which  the  fourth  term  is  1  and 
the  ninth  term  is  -g-j-g-- 

11.  Solve  the  equations 

r,   1-Sx  .  3^  +  1^      2 
^       2^2  1-3*' 

(ii.)  2x-4:7/  =  7,  Sx  +  7y=19. 

12.  A  tourist,   having  remained  behind   his  companions, 

wishes  to  rejoin  them  on  the  following  day.  He 
knows  that  they  are  5  miles  ahead,  and  that  they 
will  start  in  the  morning  at  eight  o'clock,  and  will 
walk  at  the  rate  of  3  4  miles  an  hour.  "When  must 
he  start,  in  order  to  overtake  them  at  one  o'clock, 
'  p.m.,  walking  at  the  rate  of  4  miles  an  hour,  and 
resting  once  for  half  an  .hour  on  the  road  ? 

LIV. 

Science  Schools  and  Classes,  England. 

Mathematics,  First  Stage,  May,  1880. 

Not  more  than  three  questions  are  to  be  answered.     The  number 
of  marks  assigned  to  each  question  is  given  in  brackets. 

1.  Find  the  value,  when  x  =  5  and  y  =  3,  of 

x*  -  4  x1  y  +  6  x2  y2  -  5  xy*  +  2 y4  ffi -, 

2x'  -  &x3y  +  6x2y2  -  im?  +  y*  L  "J 

2.  Multiply  a3  —  x'A  by  a2  —  x2,  and  divide  the  product  by 

(a-x)2.  [8.] 

3.  Simplify  (a-  b)(b  +  c)  (c  +  a)  +  (b  -c)  (c  +  a)  (a  +  b) 

4-  (c  —  a)  (a  +  b)  (b  +  c),  and  find  its  value  when 
0  =  1,  6=~3,  and  c  =  ~2.  [10.] 


140  EXAMINATION    PAPERS. 

4.  Solve  the  equations  "    [12.] 

(,)  7(.  +  !)-5.,(J-  +  i)  =  4.  [6.] 

(ii.)  0.5^+0.073/ =  0.93,  0.03a;-0.4y  =  0.46.    [6.] 

5.  The  rent  of  a  shop  is  -J-  of  the  rent  of  the  whole  house 

of  which  it  is  a  part.  Being  separately  rated,  its 
occupier  pays  £10  15  s.  Od.  a  year  less  in  rates  than 
the  occupier  of  the  rest  of  the  house.  The  rates  are 
3s.  Id.  in  the  pound.  What  is  the  rent  of  the 
whole  house?  [14.] 

6.  Find,  as  a  fraction  in  its  lowest  terms,  the  value  of 

1  1 


#3-3*2-15a;  +  25     xs+7xi+5x-25 


[12.] 


LV. 

Science  Schools  and  Classes,  England. 

Mathematics,  Second  Stage,  May,  1880.  —  Arithmetic  and  Algebra. 

Not  more  than'  three  questions  are  to  be  answered.     The  number 
of  marks  assigned  to  each  question  is  given  in  brackets. 

1.  Assuming  that  a  franc  is  worth  9.504c?.,  and  a  hard 

dollar  50.49  d.,  what  is  the  smallest  sum  in  francs 
that  can  be  exactly  paid  in  hard  dollars  ?  [16.] 

2.  Show  how  to  find  the  square  root  of  a  vulgar  fraction, 

so  as  to  make  sure  of  obtaining  it  in  a  finite  form,  if 
it  has  one. 
Ascertain  whether  the  square  roots  of  the  following  frac- 
tions are  finite  or  not :  jtffo  Ifflfc  2flL}  Jfa.    [18.] 

3.  Solve  two  of  the  following  sets  of  equations  :  [20.] 

(i.)   V*2  +  a% =  —?~  [10.] 

V*2+a2  a 


ALGEBRA.  141 

(ii.)  x  +  y  =7,   **+  f  =  133.       __  [10.] 

(iii.)  xyJr^/xJry  =  ll,    2xy—^JxJry  =  lo.   [10.] 

4.  What  is  the  least  integral  multiplier  which  will  make 

17a5— 68a;*y+  102  ary  ■- 68  ary  +  VJxtf  a  com- 
plete cube  ?  _  [20.] 

5.  A  rectangular  plot  of  ground  measures  42  acres,  and 

its  diagonal  is  1243  yards  long.    "What  are  its  sides  ? 

[22.] 

6.  Two  boys  start  at  the  same  instant  from  the  same  cor- 

ner of  a  square,  the  length  of  one  of  whose  sides  is 
200  yards,  and  they  run  round  it  in  opposite  direc- 
tions :  one  (A)  runs  at  the  rate  of  100  yards  in  15 
seconds,  and  loses  2  seconds  in  turning  a  corner ; 
the  other  (B)  runs  at  the  rate  of  100  yards  in  16 
seconds,  arid  loses  1  second  in  turning  a  corner. 
Where  do  they  meet?  [26.] 

LVI. 

Science  Schools  and  Classes,  Eng. 

Mathematics,  First  Stage,  May,  1881. 

Not  more  than  three  questions  are  to  be  answered.     The  number 
of  marks  assigned  to  each  question  is  given  in  brackets. 


f  ,  fz-xY      ,       z2       ,  a'  +  bW  zd    V 


1.  Show  that 

x* 

a* 

are  identical  expressions ;  that  is  to  say,  that  the  one 

can  be  deduced  from  the  other.     Find  their  values, 

when  x  — &,  z  =  4,  and  a  =  b  ==  5.  [8.] 

2.  Simplify  J  — )       f_J    [     J  ,   and  find  its  value  when 

a  =  7,  J  =  3,  c  =  2,  m  =  2,  w=l.  [8.] 


142  EXAMINATION    PAPERS. 

3.  Find  the  Q.C.M.  and  L.C.M.  of  2x*  +  9a;2  +  5  a;  -f  12 

and  2a;4  +  4a;3  +  13a;2  +11  x  +12.  [12.] 

4.  Solve  the  equations  [1-] 

(i.)  (2  +  Zx)  (i  -  g  =  (3  -  2*)  (6  -  ^ :  [0.] 

(ii.)  (x+,Jy-(x-1/f  =  3o2,x(y+b)  =  U3.        [C] 

5.  Of  two  squares  of  carpet,  one  measures  44  feet  more 

round  than  the  other,  and  187  square  feet  more  in 
area.     What  are  their  sizes?  [12.] 

6.  Oranges  are  bought  for  half-a-crown  a  hundred ;  some 

are  sold  at  3  s.  6  c?.  a  hundred,  and  the  rest  at  2  s. 
10^  d.  a  hundred.  The  same  profit  is  made,  as  if 
they  had  all  been  sold  at  3  s.  1-J-c?.  a  hundred.  Of  a 
thousand  oranges  sold,  how  many  fetch  3  s.  6d.  a 
hundred?  [14.] 

LVII. 

Science  Schools  and  Classes,  Eng. 

Mathematics,  First  Stage,  May,  1882. 

Not  more  than  three  questions  are  to  be  answered.       The  number 
of  marks  assigned  to  each  question  is  given  in  brackets. 

-1.  Divide  x3  +  8f-  27z3  +  18xyz  by  x  +  2y-3z,  and 
test  your  answer  by  substituting  x  =  5,  y  =  —  4, 
z  =  3  in  the  dividend,  divisor,  and  quotient.       [12.] 

2.  Show  that  the  product  of  1 ,  1  -| — -,  and 

.  0/        ,.      4ab  .    (x-\-a)(x  —  b) 
x4-2(a  —  o)  — is- P • 

v  J        x  x 

Write  down  the  value  of  each  factor  and  of  the  product, 
when  x  =  3  a  =  35.  •  [12.] 

3.  Reduce  to  its  lowest  terms  — — .    ,,, — —  •    [12.] 

x  —  or  —  7a;2  +  x  +  6 


ALGEBRA.  143 


4.  Divide  x2  -\-px-\-  q  by  x  —  a,  and  find  the  relation  that 

must  hold  good  between  a,  p,  and  q,  when  the  divi- 
sion can  be  performed  without  leaving  a  remainder. 

[12.] 

5.  Solve  the  equations  [12.] 

(i.)  i(0.75-*)  +  H0.47  +  2*)  =  (3-TV)*.  [«J 
(ii.)  *.j-X—"f— Wl  ["6.1 

6.  A  man  has  1000  apples  for  sale  ;  at  first  he  sells  so  as 

to  gain  at  the  rate  of  50  per  cent  on  the  cost  price ; 
when  he  has  done  this  for  a  time  the  sale  falls  off, 
so  he  sells  the  remainder  for  what  he  can  get,  and 
finds  that  by  doing  so  he  loses  at  the  rate  of  10  per 
cent;  if  his  total  gain  is  at  the  rate  of  29  per  cent, 
how  many  apples  did  he  sell  for  what  he  could  get  ? 

[14-] 

LVIII. 

Science  Schools  and  Classes,  England. 

Mathematics,  Second  Stage,  May,  1882.  —  Arithmetic  and  Algebra. 

Not  more  than  three  questions  are  to  be  answered.       The  number 
of  marks  assigned  to  each  question  is  given  in  brackets. 

1.  Given   V5  =  2.236068,  express  — ^  and  —  as 

V20  V5~+2 

decimals,  true  to  the  fifth  place.  [18.] 

2.  Solve  the  equations  [25.] 

(i.)   x*-  2a?-  3 07+  4  =  0; 
(ii.)    x2  +  y  =  8,    3^-  +  2y=-7. 

3.  A  farmer  sold  7  oxen  and  12  cows  for  £250.     He  sold 

3  more  oxen  for  £50  than  he  did  cows  for  £30. 
Required  the  price  of  each.  [25.] 


144  EXAMINATION    PAPERS." 

4.  What  is  the  term  which  must  be  added  to 

9a?4+12^3+20^  +  25  to  make  it  a  complete  square? 

t25-] 

5.  Find  (x-\-2y)\  and  obtain  the  sixth  root  of 

6,321,363,049.  [25.] 

6.  It  is  known  that  the  volume  of  a  cylinder  varies  as 

the  base  and  height  jointly.  If  the  volume  of  the 
first  of  two  cylinders  is  to  that  of  the  second  as 
11:8,  and  the  height  of  the  first  is  to  that  of  the 
second  as  3:4,  and  if  the  base  of  the  first  has  an 
area  of  16.5  square  feet,  what  is  the  area  of  the 
base  of  the  second  ?  [25.J 

LIX. 

Science  Schools  and  Classes,  England. 

Mathematics,  First  Stage,  May,  1883. 

•  Not  more  than  three  questions  are  to  be  answered.       The  number 
of  marks  assigned  to  each  question  is  given  in  brackets. 

1.  Explain  why  the  product  is  a12,  when  a5  is  multiplied 

by  a7,  and  why  the  quotient  is  a5  when  a8  is  divided 
by  a\  [8.] 

2.  Obtain  (ct*  +  3  x  +  15)2  -  (xz  —  3  x  +  15)2  in  its  simplest 

form,  and  find  its  value  when  2x  =  —  5.  [12.] 

3.  Simplify  the  expressions  [12.] 

r  s  x  —  a      a  —  b 

(1)  — x  — ; 

a      b  x 


<«>(I+eM°-9- 


Find  the  G.C.M.  of  x*  -  5xz  -  6x2  +  35^;  -  7  and 
3  xz  —  23  x2  -j-  43  x  —  8,  and  write  down  these  expres- 
sions in  factors.  [12.] 


ALGEBRA.  145 


5.  Solve  the  equations  [12.] 

W  *  0.7      ~  0.35  ' 

W   4  ""  5  "    5'    4^5      12 

6.  A  sum  of  £23  14s.  is  to  be  divided  between  A,  B,  G ; 

if  B  gets  20  per  cent  more  than  A  and  25  per  cent 
more  than  C,  how  much  does  each  get?  [16.] 

LX. 

Civil  Service  of  Great  Britain. 

Competitive  Examination  of  Candidates  for  Inspectorships  of  National 
Schools,  Ireland,  1878.  —  Time  Allowed,  3  hours. 

1.  Find  the  value  of  ,.      , when  a  =  2,    6  =  4, 

b3  —  b2-\-b  —  a 

and  the  value  of  (1  -f  a)i  X  (1  —  6)~*  when  a  =  £, 
b  =  l 

2.  Divide  6^3  +  ^2  +  a:4-2  by  %&—  x+1. 

Find  the  first  four  terms  in  the  quotient  obtained  when 
1  +  x  is  divided  by  1  —  2x. 

3.  Simplify  the  expressions 

(i.)  (16  a6  b2)h  x  (aWy  x  (2  at  5)4  ;    , 
(ii.)  il+a-\-b)3+(l+a-b)34-(l-a+bY+(l-a-b)3. 

4.  Multiply    a3  b~3  -  ab~l  +  a~1b-a,-3b3  by  ab~l  +  cr1 5, 

and  divide  (x2ymnfl  b,y  {aTf)"". 

Sx3  —  X2 07 1 

5.  Reduce  the   fraction   -— — to  its  lowest 

.     ,  3x°  —  4:x2  —  x  +  2 

terms.  , ' 

6.  Find  the  L.C.M.  of 

(i.)  7a3 bx2,  3  ax3,  6ab3; 

(ii.)  dx2Tx-2,  2x2  +  x-3. 


146  EXAMINATION    PAPERS. 

7.  Find  the  value  of  a  which  will  make  x*  —  xs  —  x2  —  ax 

divisible,  without  remainder,  by  x2  -f-  x. 

8.  Solve  the  equations 

v  ;         x     2x     3x'       K    J  1-a?       2-s 

9.  Extract  the  square  roots  of 

(i.)  9  -  4V2  ;     (ii.)  x2  -  6x  +  13  -  122T1  +  4ar2. 

10.  Solve  the  equations 

(i.)  Vx-VT+x^  =  ~; 
n  vx 

(ii.)  L-^x«z^=i; 

1  -f-  aa;      a  -}-  a; 
(iii.)  .r2  +  xy  =  32,   y2  —  xy  =  18. 

11.  A  train  is  timed  to  travel  between  two  stations,  A  and 

C,  at  45  miles  an  hour.  It  gravels  from  A  to  an 
intermediate  station,  B,  at  the  rate  of  40  miles  per 
hour,  and  the  speed  is  then  increased  to  50  miles  an 
hour.  The  train  arrives  punctually  at  the  time 
appointed.     Compare  distances  of  B  and  C  from  A. 


LXI. 

Civil  Service  of  Great/Britain. 

Open  Competitive  Examination  for  Clerkships  of  the  Superior  Class  in 
the  India  Office,  1879.  —  Time  Allowed,  3  hours. 

Full  marks  may  be   obtained  by  doing  less  than  the  whole  of 
this  paper. 

1.    If  77i  and  7i  be  positive  integers,  prove  that  amXan= am+n ; 
and,   assuming  this  formula  to  hold   good   for  all 

771 

values  of  the  indices,  deduce  the  meanings  of  o»,  a0, 
and  a~n. 


ALGEBRA.  147 

2.  Divide 

xi+yi+z4-2(x2y2+y2z2+z2x2)  by  x2+2yz-f-z2, 
and  multiply 

2a^+-Ta*yt-fl%  by  aty"1  — a%-3  +  a*3T4. 

3.  Prove  that  the  L.C.M.  of  any  two  algebraical  expres- 

sions  is   equal  to  their  product  divided   by  their 
G.C.M. 
Find  the  G.C.M.  and  L.C.M.  of  the  expressions 
(x2  +  b2)  c  +  (b2  +  c2)  x  and  (x2  -b2)c  +  (b2  -  c2)  x. 

4.  Simplify  the  following  expressions  : 

i+    ^ 

'4x2  —  6#y 
r-  \  f  CL  +  V  a-\-2x 


a2  +  2a#  —  ay  —  2xy     a2-\-  ay  —  2ax  —  2xy 
a2  -\-  2ax  -\-  ay  -\-  2xy  _ 
3  a2  +  a?/  -{-2  ax  —  2xy' 


,...  s            2  Va  —  6  .    .        3\A 

(m0  ~    / ? — ^    , ,  + 


5.  Find  the  condition  that  the  roots  of  the  equation 
ax2  -{-  bx  -\~  c  =  0  may  be  real,  and  show  that  the 
roots  of  the  equation  (x  -f  p)  (x  -\-  q)  =  pqx2  will 
always  be  real  if  p  and  q  are  real. 


6. 

Solve  the  equations 

r  n     # +  2a        3      n 

(ii.)  6^2  +  l")  +  5^  +  ^  =  38; 

(iii.)  tr3y  —  a;4  =  27,    #?/3  —  ^3y  =  84, 

148  EXAMINATION    PAPERS. 

7.  A  man,  who  drives  twice  as  fast  as  he  can  walk,  finds 

that  it  takes  him  9  hours  to  drive  to  a  certain  town 
and  to  walk  back,  and  that  when  he  has  accom- 
plished half  of  the  return  journey,  he  meets  a  man 
who  set  out  to  walk  from  the  same  place  an  hour 
and  a  half  later,  and  is  travelling  a  mile  and  a  half 
per  hour  more  slowly  than  himself.  Find  the  dis- 
tance of  the  town  from  the  starting  point,  and  the 
rate  at  which  each  man  walks. 

8.  Three  lamps  of  equal  brilliancy  are  placed  in  three 

different  corners  of  a  square  room.  Compare  the 
intensities  of  light  at  the  fourth  corner  and  at  the 
centre  of  the  room,  assuming  that  the  illumination 
from  a  source  of  light  varies  inversely  as  the  square 
of  its  distance. 

9.  Show  how  to  insert  any  number  of  harmonic  means 

between  two  given  quantities. 
If  1p  and  3  q  be  the  ^>th  and  ^th  terms,  respectively, 
of  an  harmonic  series,  prove  that  the  (p  +  ^)th  term 
will  be  6(p  —  q)> 

10.  Prove  that  the  number  of  permutations  of  n  things 

taken  r  together  is  n(n—  l)(n  —  2) (n  —  r+1). 

In  how  many  ways  can  24  ships  belonging  to  4  differ- 
ent nations  be  arranged  in  4  lines,  each  consisting  of 
6  ships  of  the  same  nationality  ? 

11.  Expand  (a2  —  x1)  ~%  to  five  terms,   and  show  that  the 

/         1  Y" 

middle  term  in  the  expansion  of  (  x  -j-  —  J  is  equal  to 

1x3x5 (2n-l)  \  ■' 

1x2x3 n 

12.  Express  the  fourth  root  of  89-28VI0  as  the  differ- 

ence of  two  surds,  and  extract  the  fifth  root  of 
99,999  correct  to  15  places  of  decimals. 


ALGEBRA. 


149 


LXII. 

Civil  Service  of  Great  Britain. 

Open  Competitive  Examination  for  Admission  to  the  Royal  Indian 
Engineering  College,  July,  1879.  —  Time  Allowed,  3  hours. 

1.  Find  the  value  of  2±$  +  b±o  +  e_±d+  d+a  when 

o  —  c      c  —  d      a  — a      a—  b 
a  =  6,  6  =  4,  c  =  3,(*=l;  and  of  a;3-4a;2+6a;~4, 
when  a;  =  1  +  V—  1. 

2.  Multiply  a?-(2+V8)y-|-(l-f-VS)2  by  s-(2- V3)y 

+  (1— V3)z;  and  divide  a:2— 4ar?/-f-?/2  by  a;  —  (2-fV3)y. 

3.  Explain  the  law  of  indices  in  the  multiplication  and 

division  of  algebraic  quantities. 
Multiply  x§  +  a$  -f  #*   by   rci  —  afi  ;  and  reduce  to  its 
simplest  form  (zk  ><:  a;"*)-*  ■+-  x~*. 

4.  Simplify  the  following  expressions  : 
gv  25a;4  +  5s*<-ar-l 


(ii.) 


20  a;4  +  a;2  - 
+ 


a  —  a;      ft  -f-  #      ft 


ft2  +  a;2 


a2 +  a* 


ft4  +  a;4 
(ft2  +  a*2)2 


5.    Solve  the  equations 

(i.)   Va^+V4+^  =  JL; 

(ii.)^_V^±iV^2a;  = 

x  +  l      \fx—\ 


(iii.) 


20 

=11; 


x—l      2\a7+l, 
(iv.)  a;2  +  y2  =  8,  2xy-tf  =  ±. 


150  EXAMINATION   PAPERS. 

6.  A  number  having  two  digits  is  to  the  number  formed 

by  inverting  the  order  of  the  digits  as  8  to  3,  and 
the  sum  of  the  two  numbers  is  99.     Find  them. 

7.  Prove  the  rule  for  finding  the  G-.C.M.  of  two  algebra- 

ical quantities. 
Find  the  L.C.M.  of 

a2-ab-\-b2,  a2  +  ab  +  b2,  az-b\  a3+b\  and  (a2-bj. 

8.  Find  the  square  roots  of 

81a;4+108^-24^  +  4  and  327-87VT5. 

9.  If  p,  q,  and  r  are  the  arithmetic,  geometric,  and  har- 

monic means  of  two  algebraical  quantities,  show  that 

(i.)  pr  =  q2 ;     (ii.)  p  -|  =  Vp2  -  q2. 

10.   Sum  to  n  terms,  and  where  possible  to  infinity, 

(i.)  3,  21    If    

(ii.)  3,  %  Iff 

1  11 


(iii.) 


2x3     3x4    4x5 


11.  Distinguish  between  permutations  and   combinations, 

and  find  the  number  of  the  latter  that  can  be  formed 
from  n  things  taken  r  together. 
In  how  many  ways  can  a  guard  of  8  soldiers  be  selected 
from  a  company  of  25,  and  in  how  many  of  them 
will  two  particular  men  be  on  guard  together  ? 

12.  Find  the  sixth  term  of  ( 3  x  —  ^  J    ,  and  the  two  middle 

terms  of  {a  —  x)9.      ^  ' 

13.  If  2_+te  =  *±^  =  £+^,  show  that 

b  -\-  cy      c-\-ay      a-\~by 
a3  +  &3+c3-3afo  =  0. 


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